Randomness and determinism in Diophantine approximation: small - - PowerPoint PPT Presentation

Randomness and determinism in Diophantine approximation: small linear forms, lattice flows and some applications

Victor Beresnevich

Department of Mathematics University of York

Imperial College London

18 January 2017

PLAN:

• V. Beresnevich (University of York)

Diophantine approximation 18 January 2017 2 / 24

PLAN:

My main goal: To have a discussion of Diophantine approximation (DA):

• V. Beresnevich (University of York)

Diophantine approximation 18 January 2017 2 / 24

PLAN:

My main goal: To have a discussion of Diophantine approximation (DA): Small linear forms, badly approximable systems

• V. Beresnevich (University of York)

Diophantine approximation 18 January 2017 2 / 24

PLAN:

My main goal: To have a discussion of Diophantine approximation (DA): Small linear forms, badly approximable systems One dimensional DA and continued fractions

• V. Beresnevich (University of York)

Diophantine approximation 18 January 2017 2 / 24

PLAN:

My main goal: To have a discussion of Diophantine approximation (DA): Small linear forms, badly approximable systems One dimensional DA and continued fractions Multiplicative DA

• V. Beresnevich (University of York)

Diophantine approximation 18 January 2017 2 / 24

PLAN:

My main goal: To have a discussion of Diophantine approximation (DA): Small linear forms, badly approximable systems One dimensional DA and continued fractions Multiplicative DA Metric DA, the Khintchine-Groshev theorem,

• V. Beresnevich (University of York)

Diophantine approximation 18 January 2017 2 / 24

PLAN:

My main goal: To have a discussion of Diophantine approximation (DA): Small linear forms, badly approximable systems One dimensional DA and continued fractions Multiplicative DA Metric DA, the Khintchine-Groshev theorem, DA on manifolds, and

• V. Beresnevich (University of York)

Diophantine approximation 18 January 2017 2 / 24

PLAN:

My main goal: To have a discussion of Diophantine approximation (DA): Small linear forms, badly approximable systems One dimensional DA and continued fractions Multiplicative DA Metric DA, the Khintchine-Groshev theorem, DA on manifolds, and Effective metric theorems

• V. Beresnevich (University of York)

Diophantine approximation 18 January 2017 2 / 24

PLAN:

My main goal: To have a discussion of Diophantine approximation (DA): Small linear forms, badly approximable systems One dimensional DA and continued fractions Multiplicative DA Metric DA, the Khintchine-Groshev theorem, DA on manifolds, and Effective metric theorems Relevance to coding and lattices:

• V. Beresnevich (University of York)

Diophantine approximation 18 January 2017 2 / 24

PLAN:

My main goal: To have a discussion of Diophantine approximation (DA): Small linear forms, badly approximable systems One dimensional DA and continued fractions Multiplicative DA Metric DA, the Khintchine-Groshev theorem, DA on manifolds, and Effective metric theorems Relevance to coding and lattices: DA is of growing interest in electronics: lattice coding and interference alignment (the latter to be briefly described);

• V. Beresnevich (University of York)

Diophantine approximation 18 January 2017 2 / 24

PLAN:

My main goal: To have a discussion of Diophantine approximation (DA): Small linear forms, badly approximable systems One dimensional DA and continued fractions Multiplicative DA Metric DA, the Khintchine-Groshev theorem, DA on manifolds, and Effective metric theorems Relevance to coding and lattices: DA is of growing interest in electronics: lattice coding and interference alignment (the latter to be briefly described); Minkowski’s theorems is a basis for much of DA, and a useful tool at the same time;

• V. Beresnevich (University of York)

Diophantine approximation 18 January 2017 2 / 24

PLAN:

My main goal: To have a discussion of Diophantine approximation (DA): Small linear forms, badly approximable systems One dimensional DA and continued fractions Multiplicative DA Metric DA, the Khintchine-Groshev theorem, DA on manifolds, and Effective metric theorems Relevance to coding and lattices: DA is of growing interest in electronics: lattice coding and interference alignment (the latter to be briefly described); Minkowski’s theorems is a basis for much of DA, and a useful tool at the same time; Dictionary between Homogeneous dynamics (lattice orbits) and Diophantine Approximation (will be discussed where possible)

• V. Beresnevich (University of York)

Diophantine approximation 18 January 2017 2 / 24

The idea of interference alignment

• V. Beresnevich (University of York)

Diophantine approximation 18 January 2017 3 / 24

The idea of interference alignment

Consider the system of linear equations:      y1 = h11x1 + · · · + h1KxK . . . . . . yB = hB1x1 + · · · + hBKxK

• V. Beresnevich (University of York)

Diophantine approximation 18 January 2017 3 / 24

The idea of interference alignment

Consider the system of linear equations:      y1 = h11x1 + · · · + h1KxK . . . . . . yB = hB1x1 + · · · + hBKxK Interpretation: hij are channel coefficients x1,. . . ,xK are signals transmitted from K different transmitters y1,. . . ,yB are signals received at a receiver B is the bandwidth (number of signaling dimensions)

• V. Beresnevich (University of York)

Diophantine approximation 18 January 2017 3 / 24

The idea of interference alignment

Consider the system of linear equations:      y1 = h11x1 + · · · + h1KxK . . . . . . yB = hB1x1 + · · · + hBKxK Interpretation: hij are channel coefficients x1,. . . ,xK are signals transmitted from K different transmitters y1,. . . ,yB are signals received at a receiver B is the bandwidth (number of signaling dimensions) If rank (hij) ≥ K the receiver can recover all the symbols x1, . . . , xK. This requires that B ≥ K.

• V. Beresnevich (University of York)

Diophantine approximation 18 January 2017 3 / 24

The idea of interference alignment

Consider the system of linear equations:      y1 = h11x1 + · · · + h1KxK . . . . . . yB = hB1x1 + · · · + hBKxK Interpretation: hij are channel coefficients x1,. . . ,xK are signals transmitted from K different transmitters y1,. . . ,yB are signals received at a receiver B is the bandwidth (number of signaling dimensions) If rank (hij) ≥ K the receiver can recover all the symbols x1, . . . , xK. This requires that B ≥ K. In IC the receiver may be interested only in some of the transmitted symbols, maybe only x1, while the other symbols are intended for other

• receivers. Even then we still require B signaling dimensions to recover x1.
• V. Beresnevich (University of York)

Diophantine approximation 18 January 2017 3 / 24

The idea of interference alignment

Consider the system of linear equations:      y1 = h11x1 + · · · + h1KxK . . . . . . yB = hB1x1 + · · · + hBKxK Interpretation: hij are channel coefficients x1,. . . ,xK are signals transmitted from K different transmitters y1,. . . ,yB are signals received at a receiver B is the bandwidth (number of signaling dimensions) If rank (hij) ≥ K the receiver can recover all the symbols x1, . . . , xK. This requires that B ≥ K. In IC the receiver may be interested only in some of the transmitted symbols, maybe only x1, while the other symbols are intended for other

• receivers. Even then we still require B signaling dimensions to recover x1.

⇒ The bandwidth is simply divided between the users.

• V. Beresnevich (University of York)

Diophantine approximation 18 January 2017 3 / 24

The idea of interference alignment

Let Hi =    h1i . . . hBi    and Y =    y1 . . . yB   

• V. Beresnevich (University of York)

Diophantine approximation 18 January 2017 4 / 24

The idea of interference alignment

Let Hi =    h1i . . . hBi    and Y =    y1 . . . yB    Then the system is Y = x1H1 + (x2H2 + · · · + xKHK)

• ∈Span {H2,...,HK }

If H1 ∈ Span {H2, . . . , HK} then x1 can be uniquely determined from the above equations.

• V. Beresnevich (University of York)

Diophantine approximation 18 January 2017 4 / 24

The idea of interference alignment

Let Hi =    h1i . . . hBi    and Y =    y1 . . . yB    Then the system is Y = x1H1 + (x2H2 + · · · + xKHK)

• ∈Span {H2,...,HK }

If H1 ∈ Span {H2, . . . , HK} then x1 can be uniquely determined from the above equations. Cadambe & Jafar (2008) showed that it is theoretically possible that each

• f the K users enjoy half of the available signalling dimensions.
• V. Beresnevich (University of York)

Diophantine approximation 18 January 2017 4 / 24

The idea of interference alignment

Let Hi =    h1i . . . hBi    and Y =    y1 . . . yB    Then the system is Y = x1H1 + (x2H2 + · · · + xKHK)

• ∈Span {H2,...,HK }

If H1 ∈ Span {H2, . . . , HK} then x1 can be uniquely determined from the above equations. Cadambe & Jafar (2008) showed that it is theoretically possible that each

• f the K users enjoy half of the available signalling dimensions. More

formally, DOF = K 2 .

• V. Beresnevich (University of York)

Diophantine approximation 18 January 2017 4 / 24

The idea of interference alignment

Let Hi =    h1i . . . hBi    and Y =    y1 . . . yB    Then the system is Y = x1H1 + (x2H2 + · · · + xKHK)

• ∈Span {H2,...,HK }

If H1 ∈ Span {H2, . . . , HK} then x1 can be uniquely determined from the above equations. Cadambe & Jafar (2008) showed that it is theoretically possible that each

• f the K users enjoy half of the available signalling dimensions. More

formally, DOF = K 2 . Various assumptions: - varying channel coefficients,

• multiple antennae at receivers
• diagonal form of the channel coefficient matrices
• . . .
• V. Beresnevich (University of York)

Diophantine approximation 18 January 2017 4 / 24

Rational dimensions framework

• V. Beresnevich (University of York)

Diophantine approximation 18 January 2017 5 / 24

Rational dimensions framework

Assume the transmit signals are taken from the constellation {0, . . . , Q}.

• V. Beresnevich (University of York)

Diophantine approximation 18 January 2017 5 / 24

Rational dimensions framework

Assume the transmit signals are taken from the constellation {0, . . . , Q}. By observing only y = h1x1 + · · · + hKxK at the receiver we still can decode our signals, if h1, . . . , hK are linearly independent over Q (or equivalently Z).

• V. Beresnevich (University of York)

Diophantine approximation 18 January 2017 5 / 24

Rational dimensions framework

Assume the transmit signals are taken from the constellation {0, . . . , Q}. By observing only y = h1x1 + · · · + hKxK at the receiver we still can decode our signals, if h1, . . . , hK are linearly independent over Q (or equivalently Z). Independence over Q means that any linear combination h1x1 + · · · + hKxK with integer x1, . . . , xK will only vanish when all the xi are zeros.

• V. Beresnevich (University of York)

Diophantine approximation 18 January 2017 5 / 24

Rational dimensions framework

Assume the transmit signals are taken from the constellation {0, . . . , Q}. By observing only y = h1x1 + · · · + hKxK at the receiver we still can decode our signals, if h1, . . . , hK are linearly independent over Q (or equivalently Z). Independence over Q means that any linear combination h1x1 + · · · + hKxK with integer x1, . . . , xK will only vanish when all the xi are zeros. If there is noise z, the receiver observes y = h1x1 + · · · + hKxK + z .

• V. Beresnevich (University of York)

Diophantine approximation 18 January 2017 5 / 24

Rational dimensions framework

Assume the transmit signals are taken from the constellation {0, . . . , Q}. By observing only y = h1x1 + · · · + hKxK at the receiver we still can decode our signals, if h1, . . . , hK are linearly independent over Q (or equivalently Z). Independence over Q means that any linear combination h1x1 + · · · + hKxK with integer x1, . . . , xK will only vanish when all the xi are zeros. If there is noise z, the receiver observes y = h1x1 + · · · + hKxK + z . This observation can be distinguished from another one, say y′ = h1x′

1 + · · · + hKx′ K + z′

• nly if y and y′ are sufficiently separated.
• V. Beresnevich (University of York)

Diophantine approximation 18 January 2017 5 / 24

Rational dimensions framework

Assume the transmit signals are taken from the constellation {0, . . . , Q}. By observing only y = h1x1 + · · · + hKxK at the receiver we still can decode our signals, if h1, . . . , hK are linearly independent over Q (or equivalently Z). Independence over Q means that any linear combination h1x1 + · · · + hKxK with integer x1, . . . , xK will only vanish when all the xi are zeros. If there is noise z, the receiver observes y = h1x1 + · · · + hKxK + z . This observation can be distinguished from another one, say y′ = h1x′

1 + · · · + hKx′ K + z′

• nly if y and y′ are sufficiently separated.

Fact: for any ε > 0 for almost every collection (h1, . . . , hK) of real numbers there exists a constant γ = γ(ε, h1, . . . , hK) > 0 such that |h1(x1 − x′

1) + · · · + hK(xK − x′ K)| ≥

γ QK−1+ε whenever (x1, . . . , xK) = (x′

1, . . . , x′ K).

• V. Beresnevich (University of York)

Diophantine approximation 18 January 2017 5 / 24

Rational dimensions: single antenna, multiple data streams

Using that Fact a single antenna can be turned into a multiple antennae to simultaneously transmit several data streams, say x1, . . . , xM, xi ∈ {0, . . . , Q}.

• V. Beresnevich (University of York)

Diophantine approximation 18 January 2017 6 / 24

Rational dimensions: single antenna, multiple data streams

Using that Fact a single antenna can be turned into a multiple antennae to simultaneously transmit several data streams, say x1, . . . , xM, xi ∈ {0, . . . , Q}. The transmit signal may be taken to be x = λ(c1x1 + · · · + cMxM) where |ci| = O(1) are precoding coefficients and λ is a scalar reflecting power constraints.

• V. Beresnevich (University of York)

Diophantine approximation 18 January 2017 6 / 24

Rational dimensions: single antenna, multiple data streams

Using that Fact a single antenna can be turned into a multiple antennae to simultaneously transmit several data streams, say x1, . . . , xM, xi ∈ {0, . . . , Q}. The transmit signal may be taken to be x = λ(c1x1 + · · · + cMxM) where |ci| = O(1) are precoding coefficients and λ is a scalar reflecting power constraints. The separation between different x’s is at least λO(Q−M+1−ε).

• V. Beresnevich (University of York)

Diophantine approximation 18 January 2017 6 / 24

Rational dimensions: single antenna, multiple data streams

Using that Fact a single antenna can be turned into a multiple antennae to simultaneously transmit several data streams, say x1, . . . , xM, xi ∈ {0, . . . , Q}. The transmit signal may be taken to be x = λ(c1x1 + · · · + cMxM) where |ci| = O(1) are precoding coefficients and λ is a scalar reflecting power constraints. The separation between different x’s is at least λO(Q−M+1−ε). Choosing Q = O(P

1−ε 2(M+ε) )

and λ = P1/2Q−1 ensures that the separation is bigger than the standard deviation of the noise, while meeting power constraints.

• V. Beresnevich (University of York)

Diophantine approximation 18 January 2017 6 / 24

Rational dimensions: single antenna, multiple data streams

Using that Fact a single antenna can be turned into a multiple antennae to simultaneously transmit several data streams, say x1, . . . , xM, xi ∈ {0, . . . , Q}. The transmit signal may be taken to be x = λ(c1x1 + · · · + cMxM) where |ci| = O(1) are precoding coefficients and λ is a scalar reflecting power constraints. The separation between different x’s is at least λO(Q−M+1−ε). Choosing Q = O(P

1−ε 2(M+ε) )

and λ = P1/2Q−1 ensures that the separation is bigger than the standard deviation of the noise, while meeting power constraints. More sophisticated examples require separability of linear forms when the coefficients the forms are functions of several other variables (in DA this is known as Diophantine approximation on manifolds).

• V. Beresnevich (University of York)

Diophantine approximation 18 January 2017 6 / 24

Rational dimensions: single antenna, multiple data streams

Using that Fact a single antenna can be turned into a multiple antennae to simultaneously transmit several data streams, say x1, . . . , xM, xi ∈ {0, . . . , Q}. The transmit signal may be taken to be x = λ(c1x1 + · · · + cMxM) where |ci| = O(1) are precoding coefficients and λ is a scalar reflecting power constraints. The separation between different x’s is at least λO(Q−M+1−ε). Choosing Q = O(P

1−ε 2(M+ε) )

and λ = P1/2Q−1 ensures that the separation is bigger than the standard deviation of the noise, while meeting power constraints. More sophisticated examples require separability of linear forms when the coefficients the forms are functions of several other variables (in DA this is known as Diophantine approximation on manifolds). Just as with vector alignment, it is possible to align alone real numbers.

• V. Beresnevich (University of York)

Diophantine approximation 18 January 2017 6 / 24

2-user X-channel (Motahari, et al)

• V. Beresnevich (University of York)

Diophantine approximation 18 January 2017 7 / 24

2-user X-channel (Motahari, et al)

x1 y1 x2 y2

• ✒
• ❅

❅ ❅ ❅ ❅ ❅ ❅ ❅ ❘ ✲ ✲

T1 T2 R1

u1, u2?

R2

v1, v2?

h11 h22 h12 h21

u1, v1 u2, v2

• V. Beresnevich (University of York)

Diophantine approximation 18 January 2017 7 / 24

2-user X-channel (Motahari, et al)

x1 y1 x2 y2

• ✒
• ❅

❅ ❅ ❅ ❅ ❅ ❅ ❅ ❘ ✲ ✲

T1 T2 R1

u1, u2?

R2

v1, v2?

h11 h22 h12 h21

u1, v1 u2, v2 T1 simultaneously transmits two data streams u1 (intended for R1) and v1 (intended for R2). Similarly, T2 transmits independent two data streams u2 (intended for R1) and v2 (intended for R2). hij are the the channel

• coefficients. Let xi is the signal transmitted by Ti.
• V. Beresnevich (University of York)

Diophantine approximation 18 January 2017 7 / 24

2-user X-channel (Motahari, et al)

x1 y1 x2 y2

• ✒
• ❅

❅ ❅ ❅ ❅ ❅ ❅ ❅ ❘ ✲ ✲

T1 T2 R1

u1, u2?

R2

v1, v2?

h11 h22 h12 h21

u1, v1 u2, v2 T1 simultaneously transmits two data streams u1 (intended for R1) and v1 (intended for R2). Similarly, T2 transmits independent two data streams u2 (intended for R1) and v2 (intended for R2). hij are the the channel

• coefficients. Let xi is the signal transmitted by Ti. The received signals by

R1 and R2 are y1 = h11x1 + h12x2 + z1 , y2 = h21x1 + h22x2 + z2 .

• V. Beresnevich (University of York)

Diophantine approximation 18 January 2017 7 / 24

2-user X-channel (Motahari, et al)

y1 = h11x1 + h12x2 + z1 , y2 = h21x1 + h22x2 + z2 .

• V. Beresnevich (University of York)

Diophantine approximation 18 January 2017 8 / 24

2-user X-channel (Motahari, et al)

y1 = h11x1 + h12x2 + z1 , y2 = h21x1 + h22x2 + z2 . If x1 = λ(h22u1 + h12v1) and x2 = λ(h21u2 + h11v2) then y1 = λ

• h11h22u1 + h12h21u2 + h11h12(v1 + v2)
• + z1,

y2 = λ

• h21h22(u1 + u2) + h12h21v1 + h11h22v2
• + z1

At R1: v1 and v2 are aligned along the same real number, h11h12 At R2: u1 and u2 are aligned along the same real number, h21h22

• V. Beresnevich (University of York)

Diophantine approximation 18 January 2017 8 / 24

2-user X-channel (Motahari, et al)

y1 = h11x1 + h12x2 + z1 , y2 = h21x1 + h22x2 + z2 . If x1 = λ(h22u1 + h12v1) and x2 = λ(h21u2 + h11v2) then y1 = λ

• h11h22u1 + h12h21u2 + h11h12(v1 + v2)
• + z1,

y2 = λ

• h21h22(u1 + u2) + h12h21v1 + h11h22v2
• + z1

At R1: v1 and v2 are aligned along the same real number, h11h12 At R2: u1 and u2 are aligned along the same real number, h21h22 Again, assuming ui, vi ∈ {0, . . . , Q} we achieve the required separation (and normalising power) in each of the equation by taking Q = O(P

1−ε 2(3+ε) ),

λ = O(P

1+ε 3+ε ) .

DOF= 4

3 almost surely.

• V. Beresnevich (University of York)

Diophantine approximation 18 January 2017 8 / 24

Diophantine approximation: the basics

In applications we often deal with small linear forms or systems of small linear forms at integer variables -

• V. Beresnevich (University of York)

Diophantine approximation 18 January 2017 9 / 24

Diophantine approximation: the basics

In applications we often deal with small linear forms or systems of small linear forms at integer variables - a subject of Diophantine approximation.

• V. Beresnevich (University of York)

Diophantine approximation 18 January 2017 9 / 24

Diophantine approximation: the basics

In applications we often deal with small linear forms or systems of small linear forms at integer variables - a subject of Diophantine approximation. Dirichlet’s theorem: Let αi,j ∈ R, where 1 ≤ j ≤ n, 1 ≤ i ≤ m, and Q > 1. Then there exist q1, . . . , qn, p1, . . . , pm ∈ Z such that |q1αi,1 + · · · + qnαi,n − pi| < Q− n

m

(1 ≤ i ≤ m) 1 ≤ max

1≤j≤n |qj| ≤ Q .

• V. Beresnevich (University of York)

Diophantine approximation 18 January 2017 9 / 24

Diophantine approximation: the basics

In applications we often deal with small linear forms or systems of small linear forms at integer variables - a subject of Diophantine approximation. Dirichlet’s theorem: Let αi,j ∈ R, where 1 ≤ j ≤ n, 1 ≤ i ≤ m, and Q > 1. Then there exist q1, . . . , qn, p1, . . . , pm ∈ Z such that |q1αi,1 + · · · + qnαi,n − pi| < Q− n

m

(1 ≤ i ≤ m) 1 ≤ max

1≤j≤n |qj| ≤ Q .

Minkowski’s theorem for systems of linear forms: Let βi,j ∈ R, where 1 ≤ i, j ≤ k, and let C1, . . . , Ck > 0. If | det(βi,j)1≤i,j≤k| ≤

k

• i=1

Ci, (1) then there exist a non-zero integer point x = (x1, . . . , xk) such that

• |x1βi,1 + · · · + xkβi,k| < Ci ,

(1 ≤ i ≤ k − 1) |x1βk,1 + · · · + xnβk,k| ≤ Ck . (2)

• V. Beresnevich (University of York)

Diophantine approximation 18 January 2017 9 / 24

Diophantine approximation: the basics

In applications we often deal with small linear forms or systems of small linear forms at integer variables - a subject of Diophantine approximation. Dirichlet’s theorem: Let αi,j ∈ R, where 1 ≤ j ≤ n, 1 ≤ i ≤ m, and Q > 1. Then there exist q1, . . . , qn, p1, . . . , pm ∈ Z such that |q1αi,1 + · · · + qnαi,n − pi| < Q− n

m

(1 ≤ i ≤ m) 1 ≤ max

1≤j≤n |qj| ≤ Q .

Minkowski’s theorem for systems of linear forms: Let βi,j ∈ R, where 1 ≤ i, j ≤ k, and let C1, . . . , Ck > 0. If | det(βi,j)1≤i,j≤k| ≤

k

• i=1

Ci, (1) then there exist a non-zero integer point x = (x1, . . . , xk) such that

• |x1βi,1 + · · · + xkβi,k| < Ci ,

(1 ≤ i ≤ k − 1) |x1βk,1 + · · · + xnβk,k| ≤ Ck . (2) Proof: uses Minkowski’s theorem for convex bodies.

• V. Beresnevich (University of York)

Diophantine approximation 18 January 2017 9 / 24

Diophantine approximation: the basics

Minkowski’s theorem: If B is a convex body in Rn symmetric about 0 and vol B > 2n then B contains a non-zero integer point.

• V. Beresnevich (University of York)

Diophantine approximation 18 January 2017 10 / 24

Diophantine approximation: the basics

Minkowski’s theorem: If B is a convex body in Rn symmetric about 0 and vol B > 2n then B contains a non-zero integer point. Minkowski can be used beyond the real case. Example: Dirichlet’s theorem for C (one dimensional): For any z ∈ C and any Q > 1 there exist p, q ∈ Z[i] = {a + bi : a, b ∈ Z} (i = √−1) such that |qz − p| < 4 πQ , 1 ≤ |q| ≤ Q .

• V. Beresnevich (University of York)

Diophantine approximation 18 January 2017 10 / 24

Diophantine approximation: the basics

Minkowski’s theorem: If B is a convex body in Rn symmetric about 0 and vol B > 2n then B contains a non-zero integer point. Minkowski can be used beyond the real case. Example: Dirichlet’s theorem for C (one dimensional): For any z ∈ C and any Q > 1 there exist p, q ∈ Z[i] = {a + bi : a, b ∈ Z} (i = √−1) such that |qz − p| < 4 πQ , 1 ≤ |q| ≤ Q . Badly approximable systems/matrices: Let αi,j ∈ R, where 1 ≤ j ≤ n, 1 ≤ i ≤ m, and Q > 1. Then A = (αi,j)i,j is badly approximable if there exist c > 0 such that for all Q > 1 the only integer solution (q1, . . . , qn, p1, . . . , pm) to the system |q1αi,1 + · · · + qnαi,n − pi| < cQ− n

m

(1 ≤ i ≤ m) |qi| ≤ Q (1 ≤ j ≤ n) is zero.

• V. Beresnevich (University of York)

Diophantine approximation 18 January 2017 10 / 24

Diophantine approximation: the basics

Minkowski’s theorem: If B is a convex body in Rn symmetric about 0 and vol B > 2n then B contains a non-zero integer point. Minkowski can be used beyond the real case. Example: Dirichlet’s theorem for C (one dimensional): For any z ∈ C and any Q > 1 there exist p, q ∈ Z[i] = {a + bi : a, b ∈ Z} (i = √−1) such that |qz − p| < 4 πQ , 1 ≤ |q| ≤ Q . Badly approximable systems/matrices: Let αi,j ∈ R, where 1 ≤ j ≤ n, 1 ≤ i ≤ m, and Q > 1. Then A = (αi,j)i,j is badly approximable if there exist c > 0 such that for all Q > 1 the only integer solution (q1, . . . , qn, p1, . . . , pm) to the system |q1αi,1 + · · · + qnαi,n − pi| < cQ− n

m

(1 ≤ i ≤ m) |qi| ≤ Q (1 ≤ j ≤ n) is zero. Bad(n, m) is the set of badly approximable m × n matrices.

• V. Beresnevich (University of York)

Diophantine approximation 18 January 2017 10 / 24

Weighted Diophantine approximation

Dirichlet’s theorem with weights: Let αi,j ∈ R, where 1 ≤ j ≤ n, 1 ≤ i ≤ m. Let c = 1 and let r = (r1, . . . , rn), s = (s1, . . . , sm) be such that sj ≥ 0, ri ≥ 0, s1 + · · · + sm = 1, r1 + · · · + rn = 1 . Then for any Q > 1 there exist q1, . . . , qn, p1, . . . , pm ∈ Z such that |q1αi,1 + · · · + qnαi,n − pi| < c Q−si (1 ≤ i ≤ m) |qj| ≤ Qrj (1 ≤ j ≤ n) (q1, . . . , qn) = 0 .

• V. Beresnevich (University of York)

Diophantine approximation 18 January 2017 11 / 24

Weighted Diophantine approximation

Dirichlet’s theorem with weights: Let αi,j ∈ R, where 1 ≤ j ≤ n, 1 ≤ i ≤ m. Let c = 1 and let r = (r1, . . . , rn), s = (s1, . . . , sm) be such that sj ≥ 0, ri ≥ 0, s1 + · · · + sm = 1, r1 + · · · + rn = 1 . Then for any Q > 1 there exist q1, . . . , qn, p1, . . . , pm ∈ Z such that |q1αi,1 + · · · + qnαi,n − pi| < c Q−si (1 ≤ i ≤ m) |qj| ≤ Qrj (1 ≤ j ≤ n) (q1, . . . , qn) = 0 . Weighted Badly approximable systems/matrices: Let αi,j ∈ R, where 1 ≤ j ≤ n, 1 ≤ i ≤ m, and Q > 1. Then A = (αi,j)i,j is (r, s)-badly approximable if there exist c > 0 such that for all Q > 1 the only integer solution (q1, . . . , qn, p1, . . . , pm) to the above system is zero.

• V. Beresnevich (University of York)

Diophantine approximation 18 January 2017 11 / 24

Weighted Diophantine approximation

Dirichlet’s theorem with weights: Let αi,j ∈ R, where 1 ≤ j ≤ n, 1 ≤ i ≤ m. Let c = 1 and let r = (r1, . . . , rn), s = (s1, . . . , sm) be such that sj ≥ 0, ri ≥ 0, s1 + · · · + sm = 1, r1 + · · · + rn = 1 . Then for any Q > 1 there exist q1, . . . , qn, p1, . . . , pm ∈ Z such that |q1αi,1 + · · · + qnαi,n − pi| < c Q−si (1 ≤ i ≤ m) |qj| ≤ Qrj (1 ≤ j ≤ n) (q1, . . . , qn) = 0 . Weighted Badly approximable systems/matrices: Let αi,j ∈ R, where 1 ≤ j ≤ n, 1 ≤ i ≤ m, and Q > 1. Then A = (αi,j)i,j is (r, s)-badly approximable if there exist c > 0 such that for all Q > 1 the only integer solution (q1, . . . , qn, p1, . . . , pm) to the above system is zero. Let Bad(r, s) be the set of (r, s)-badly approximable m × n matrices.

• V. Beresnevich (University of York)

Diophantine approximation 18 January 2017 11 / 24

Weighted Diophantine approximation

Dirichlet’s theorem with weights: Let αi,j ∈ R, where 1 ≤ j ≤ n, 1 ≤ i ≤ m. Let c = 1 and let r = (r1, . . . , rn), s = (s1, . . . , sm) be such that sj ≥ 0, ri ≥ 0, s1 + · · · + sm = 1, r1 + · · · + rn = 1 . Then for any Q > 1 there exist q1, . . . , qn, p1, . . . , pm ∈ Z such that |q1αi,1 + · · · + qnαi,n − pi| < c Q−si (1 ≤ i ≤ m) |qj| ≤ Qrj (1 ≤ j ≤ n) (q1, . . . , qn) = 0 . Weighted Badly approximable systems/matrices: Let αi,j ∈ R, where 1 ≤ j ≤ n, 1 ≤ i ≤ m, and Q > 1. Then A = (αi,j)i,j is (r, s)-badly approximable if there exist c > 0 such that for all Q > 1 the only integer solution (q1, . . . , qn, p1, . . . , pm) to the above system is zero. Let Bad(r, s) be the set of (r, s)-badly approximable m × n matrices. Bad(n, m) = Bad(( 1

n, . . . , 1 n), ( 1 m, . . . , 1 m)).

• V. Beresnevich (University of York)

Diophantine approximation 18 January 2017 11 / 24

Weighted Diophantine approximation

Dirichlet’s theorem with weights: Let αi,j ∈ R, where 1 ≤ j ≤ n, 1 ≤ i ≤ m. Let c = 1 and let r = (r1, . . . , rn), s = (s1, . . . , sm) be such that sj ≥ 0, ri ≥ 0, s1 + · · · + sm = 1, r1 + · · · + rn = 1 . Then for any Q > 1 there exist q1, . . . , qn, p1, . . . , pm ∈ Z such that |q1αi,1 + · · · + qnαi,n − pi| < c Q−si (1 ≤ i ≤ m) |qj| ≤ Qrj (1 ≤ j ≤ n) (q1, . . . , qn) = 0 . Weighted Badly approximable systems/matrices: Let αi,j ∈ R, where 1 ≤ j ≤ n, 1 ≤ i ≤ m, and Q > 1. Then A = (αi,j)i,j is (r, s)-badly approximable if there exist c > 0 such that for all Q > 1 the only integer solution (q1, . . . , qn, p1, . . . , pm) to the above system is zero. Let Bad(r, s) be the set of (r, s)-badly approximable m × n matrices. Bad(n, m) = Bad(( 1

n, . . . , 1 n), ( 1 m, . . . , 1 m)).

The transpose of Bad(r, s) is Bad(s, r).

• V. Beresnevich (University of York)

Diophantine approximation 18 January 2017 11 / 24

Weighted Diophantine approximation

Dirichlet’s theorem with weights: Let αi,j ∈ R, where 1 ≤ j ≤ n, 1 ≤ i ≤ m. Let c = 1 and let r = (r1, . . . , rn), s = (s1, . . . , sm) be such that sj ≥ 0, ri ≥ 0, s1 + · · · + sm = 1, r1 + · · · + rn = 1 . Then for any Q > 1 there exist q1, . . . , qn, p1, . . . , pm ∈ Z such that |q1αi,1 + · · · + qnαi,n − pi| < c Q−si (1 ≤ i ≤ m) |qj| ≤ Qrj (1 ≤ j ≤ n) (q1, . . . , qn) = 0 . Weighted Badly approximable systems/matrices: Let αi,j ∈ R, where 1 ≤ j ≤ n, 1 ≤ i ≤ m, and Q > 1. Then A = (αi,j)i,j is (r, s)-badly approximable if there exist c > 0 such that for all Q > 1 the only integer solution (q1, . . . , qn, p1, . . . , pm) to the above system is zero. Let Bad(r, s) be the set of (r, s)-badly approximable m × n matrices. Bad(n, m) = Bad(( 1

n, . . . , 1 n), ( 1 m, . . . , 1 m)).

The transpose of Bad(r, s) is Bad(s, r). The transpose of Bad(n, m) equals Bad(m, n).

• V. Beresnevich (University of York)

Diophantine approximation 18 January 2017 11 / 24

Multiplicative Diophantine approximation

• V. Beresnevich (University of York)

Diophantine approximation 18 January 2017 12 / 24

Multiplicative Diophantine approximation

Littlewood’s conjecture (LC), (1930): For any α1, α2 ∈ R and any ε > 0 there exists q ∈ Z=0, p1, p2 ∈ Z such that |q| · |qα1 − p1| · |qα2 − p2| < ε .

• V. Beresnevich (University of York)

Diophantine approximation 18 January 2017 12 / 24

Multiplicative Diophantine approximation

Littlewood’s conjecture (LC), (1930): For any α1, α2 ∈ R and any ε > 0 there exists q ∈ Z=0, p1, p2 ∈ Z such that |q| · |qα1 − p1| · |qα2 − p2| < ε . What’s known:

• V. Beresnevich (University of York)

Diophantine approximation 18 January 2017 12 / 24

Multiplicative Diophantine approximation

Littlewood’s conjecture (LC), (1930): For any α1, α2 ∈ R and any ε > 0 there exists q ∈ Z=0, p1, p2 ∈ Z such that |q| · |qα1 − p1| · |qα2 − p2| < ε . What’s known:

• LC holds if either α1 or α2 is not badly approximable.
• V. Beresnevich (University of York)

Diophantine approximation 18 January 2017 12 / 24

Multiplicative Diophantine approximation

Littlewood’s conjecture (LC), (1930): For any α1, α2 ∈ R and any ε > 0 there exists q ∈ Z=0, p1, p2 ∈ Z such that |q| · |qα1 − p1| · |qα2 − p2| < ε . What’s known:

• LC holds if either α1 or α2 is not badly approximable.
• LC holds if α1 and α2 lie in the same cubic field (Cassels &

Swinnerton-Dyer, 1955)

• V. Beresnevich (University of York)

Diophantine approximation 18 January 2017 12 / 24

Multiplicative Diophantine approximation

Littlewood’s conjecture (LC), (1930): For any α1, α2 ∈ R and any ε > 0 there exists q ∈ Z=0, p1, p2 ∈ Z such that |q| · |qα1 − p1| · |qα2 − p2| < ε . What’s known:

• LC holds if either α1 or α2 is not badly approximable.
• LC holds if α1 and α2 lie in the same cubic field (Cassels &

Swinnerton-Dyer, 1955)

• The set of counterexample to LC has zero dimension (Einsiedler, Katok

and Lindenstrauss, 2006), led Lindenstrauss to win a Fields medal in 2010

• V. Beresnevich (University of York)

Diophantine approximation 18 January 2017 12 / 24

Multiplicative Diophantine approximation

Littlewood’s conjecture (LC), (1930): For any α1, α2 ∈ R and any ε > 0 there exists q ∈ Z=0, p1, p2 ∈ Z such that |q| · |qα1 − p1| · |qα2 − p2| < ε . What’s known:

• LC holds if either α1 or α2 is not badly approximable.
• LC holds if α1 and α2 lie in the same cubic field (Cassels &

Swinnerton-Dyer, 1955)

• The set of counterexample to LC has zero dimension (Einsiedler, Katok

and Lindenstrauss, 2006), led Lindenstrauss to win a Fields medal in 2010

• It is not known if LC holds for (

√ 2, √ 3) or any other explicit examples

• f badly approximable pairs.
• V. Beresnevich (University of York)

Diophantine approximation 18 January 2017 12 / 24

Multiplicative Diophantine approximation

Littlewood’s conjecture (LC), (1930): For any α1, α2 ∈ R and any ε > 0 there exists q ∈ Z=0, p1, p2 ∈ Z such that |q| · |qα1 − p1| · |qα2 − p2| < ε .

• V. Beresnevich (University of York)

Diophantine approximation 18 January 2017 13 / 24

Multiplicative Diophantine approximation

Littlewood’s conjecture (LC), (1930): For any α1, α2 ∈ R and any ε > 0 there exists q ∈ Z=0, p1, p2 ∈ Z such that |q| · |qα1 − p1| · |qα2 − p2| < ε . Let Λ =   1 α1 1 α2 1   Z3 = {   qα1 − p qα2 − p q   : q, p1, p2 ∈ Z} .

• V. Beresnevich (University of York)

Diophantine approximation 18 January 2017 13 / 24

Multiplicative Diophantine approximation

Littlewood’s conjecture (LC), (1930): For any α1, α2 ∈ R and any ε > 0 there exists q ∈ Z=0, p1, p2 ∈ Z such that |q| · |qα1 − p1| · |qα2 − p2| < ε . Let Λ =   1 α1 1 α2 1   Z3 = {   qα1 − p qα2 − p q   : q, p1, p2 ∈ Z} . LC: inf

(λ1,λ2,λ3)∈Λ, λ3=0 |λ1λ2λ3| = 0 .

• V. Beresnevich (University of York)

Diophantine approximation 18 January 2017 13 / 24

Multiplicative Diophantine approximation

Littlewood’s conjecture (LC), (1930): For any α1, α2 ∈ R and any ε > 0 there exists q ∈ Z=0, p1, p2 ∈ Z such that |q| · |qα1 − p1| · |qα2 − p2| < ε . Let Λ =   1 α1 1 α2 1   Z3 = {   qα1 − p qα2 − p q   : q, p1, p2 ∈ Z} . LC: inf

(λ1,λ2,λ3)∈Λ, λ3=0 |λ1λ2λ3| = 0 .

Admissible lattices: Λ ∈ Ln := GL(n, R)/SL(n, Z) is admissible if Nm(Λ) := inf{|λ1 · · · λn| : (λ1, . . . , λn) ∈ Λ=0} > 0 .

• V. Beresnevich (University of York)

Diophantine approximation 18 January 2017 13 / 24

Multiplicative Diophantine approximation

Littlewood’s conjecture (LC), (1930): For any α1, α2 ∈ R and any ε > 0 there exists q ∈ Z=0, p1, p2 ∈ Z such that |q| · |qα1 − p1| · |qα2 − p2| < ε . Let Λ =   1 α1 1 α2 1   Z3 = {   qα1 − p qα2 − p q   : q, p1, p2 ∈ Z} . LC: inf

(λ1,λ2,λ3)∈Λ, λ3=0 |λ1λ2λ3| = 0 .

Admissible lattices: Λ ∈ Ln := GL(n, R)/SL(n, Z) is admissible if Nm(Λ) := inf{|λ1 · · · λn| : (λ1, . . . , λn) ∈ Λ=0} > 0 . Fact: The set of admissible lattices is dense in Ln.

• V. Beresnevich (University of York)

Diophantine approximation 18 January 2017 13 / 24

Multiplicative Diophantine approximation

Littlewood’s conjecture (LC), (1930): For any α1, α2 ∈ R and any ε > 0 there exists q ∈ Z=0, p1, p2 ∈ Z such that |q| · |qα1 − p1| · |qα2 − p2| < ε . Let Λ =   1 α1 1 α2 1   Z3 = {   qα1 − p qα2 − p q   : q, p1, p2 ∈ Z} . LC: inf

(λ1,λ2,λ3)∈Λ, λ3=0 |λ1λ2λ3| = 0 .

Admissible lattices: Λ ∈ Ln := GL(n, R)/SL(n, Z) is admissible if Nm(Λ) := inf{|λ1 · · · λn| : (λ1, . . . , λn) ∈ Λ=0} > 0 . Fact: The set of admissible lattices is dense in Ln. n = 2:

• V. Beresnevich (University of York)

Diophantine approximation 18 January 2017 13 / 24

Multiplicative Diophantine approximation

Littlewood’s conjecture (LC), (1930): For any α1, α2 ∈ R and any ε > 0 there exists q ∈ Z=0, p1, p2 ∈ Z such that |q| · |qα1 − p1| · |qα2 − p2| < ε . Let Λ =   1 α1 1 α2 1   Z3 = {   qα1 − p qα2 − p q   : q, p1, p2 ∈ Z} . LC: inf

(λ1,λ2,λ3)∈Λ, λ3=0 |λ1λ2λ3| = 0 .

Admissible lattices: Λ ∈ Ln := GL(n, R)/SL(n, Z) is admissible if Nm(Λ) := inf{|λ1 · · · λn| : (λ1, . . . , λn) ∈ Λ=0} > 0 . Fact: The set of admissible lattices is dense in Ln. n = 2: Λ = a b c d

• Z2 is admissible
• V. Beresnevich (University of York)

Diophantine approximation 18 January 2017 13 / 24

Multiplicative Diophantine approximation

Littlewood’s conjecture (LC), (1930): For any α1, α2 ∈ R and any ε > 0 there exists q ∈ Z=0, p1, p2 ∈ Z such that |q| · |qα1 − p1| · |qα2 − p2| < ε . Let Λ =   1 α1 1 α2 1   Z3 = {   qα1 − p qα2 − p q   : q, p1, p2 ∈ Z} . LC: inf

(λ1,λ2,λ3)∈Λ, λ3=0 |λ1λ2λ3| = 0 .

Admissible lattices: Λ ∈ Ln := GL(n, R)/SL(n, Z) is admissible if Nm(Λ) := inf{|λ1 · · · λn| : (λ1, . . . , λn) ∈ Λ=0} > 0 . Fact: The set of admissible lattices is dense in Ln. n = 2: Λ = a b c d

• Z2 is admissible ⇐

⇒

• V. Beresnevich (University of York)

Diophantine approximation 18 January 2017 13 / 24

Multiplicative Diophantine approximation

Littlewood’s conjecture (LC), (1930): For any α1, α2 ∈ R and any ε > 0 there exists q ∈ Z=0, p1, p2 ∈ Z such that |q| · |qα1 − p1| · |qα2 − p2| < ε . Let Λ =   1 α1 1 α2 1   Z3 = {   qα1 − p qα2 − p q   : q, p1, p2 ∈ Z} . LC: inf

(λ1,λ2,λ3)∈Λ, λ3=0 |λ1λ2λ3| = 0 .

Admissible lattices: Λ ∈ Ln := GL(n, R)/SL(n, Z) is admissible if Nm(Λ) := inf{|λ1 · · · λn| : (λ1, . . . , λn) ∈ Λ=0} > 0 . Fact: The set of admissible lattices is dense in Ln. n = 2: Λ = a b c d

• Z2 is admissible ⇐

⇒ a b, c d ∈ Bad = Bad(1, 1) ⊂ R.

• V. Beresnevich (University of York)

Diophantine approximation 18 January 2017 13 / 24

Admissible lattices: examples

Admissible lattices: Λ ∈ Ln := GL(n, R)/SL(n, Z) is admissible if Nm(Λ) := inf{|λ1 · · · λn| : (λ1, . . . , λn) ∈ Λ=0} > 0 . n = 2: Λ = a b c d

• Z2 is admissible ⇐

⇒ a b, c d ∈ Bad = Bad(1, 1) ⊂ R. Example (admissible lattices in Ln for n ≥ 3):

• V. Beresnevich (University of York)

Diophantine approximation 18 January 2017 14 / 24

Admissible lattices: examples

Admissible lattices: Λ ∈ Ln := GL(n, R)/SL(n, Z) is admissible if Nm(Λ) := inf{|λ1 · · · λn| : (λ1, . . . , λn) ∈ Λ=0} > 0 . n = 2: Λ = a b c d

• Z2 is admissible ⇐

⇒ a b, c d ∈ Bad = Bad(1, 1) ⊂ R. Example (admissible lattices in Ln for n ≥ 3): Let f ∈ Z[x], monic, irreducible over Q, deg f = n with all real roots. Thus f (x) = (x − α1) . . . (x − αn) with αi ∈ R.

• V. Beresnevich (University of York)

Diophantine approximation 18 January 2017 14 / 24

Admissible lattices: examples

Admissible lattices: Λ ∈ Ln := GL(n, R)/SL(n, Z) is admissible if Nm(Λ) := inf{|λ1 · · · λn| : (λ1, . . . , λn) ∈ Λ=0} > 0 . n = 2: Λ = a b c d

• Z2 is admissible ⇐

⇒ a b, c d ∈ Bad = Bad(1, 1) ⊂ R. Example (admissible lattices in Ln for n ≥ 3): Let f ∈ Z[x], monic, irreducible over Q, deg f = n with all real roots. Thus f (x) = (x − α1) . . . (x − αn) with αi ∈ R. Define Λ =    1 α1 . . . αn−1

1

. . . ... . . . 1 αn . . . αn−1

n

   Zn .

• V. Beresnevich (University of York)

Diophantine approximation 18 January 2017 14 / 24

Admissible lattices: examples

Admissible lattices: Λ ∈ Ln := GL(n, R)/SL(n, Z) is admissible if Nm(Λ) := inf{|λ1 · · · λn| : (λ1, . . . , λn) ∈ Λ=0} > 0 . n = 2: Λ = a b c d

• Z2 is admissible ⇐

⇒ a b, c d ∈ Bad = Bad(1, 1) ⊂ R. Example (admissible lattices in Ln for n ≥ 3): Let f ∈ Z[x], monic, irreducible over Q, deg f = n with all real roots. Thus f (x) = (x − α1) . . . (x − αn) with αi ∈ R. Define Λ =    1 α1 . . . αn−1

1

. . . ... . . . 1 αn . . . αn−1

n

   Zn . Nm(Λ) = inf

1≤i≤n

|g(αi)| : g ∈ Z[x]=0, deg g ≤ n − 1

• = inf
• |Resultant(f , g)| : g ∈ Z[x]=0, deg g ≤ n − 1
• ≥ 1 .
• V. Beresnevich (University of York)

Diophantine approximation 18 January 2017 14 / 24

Admissible lattices and Bad

• V. Beresnevich (University of York)

Diophantine approximation 18 January 2017 15 / 24

Admissible lattices and Bad

Let g be an (n + 1) × (n + 1) matrix such that Λ = gZn+1 is admissible. Let (x0, x1, . . . , xn) be any row (column) of g.

• V. Beresnevich (University of York)

Diophantine approximation 18 January 2017 15 / 24

Admissible lattices and Bad

Let g be an (n + 1) × (n + 1) matrix such that Λ = gZn+1 is admissible. Let (x0, x1, . . . , xn) be any row (column) of g. Then x0 = 0 and x1 x0 , . . . , xn x0

• ∈ Bad(n, 1) .

Proof.

• V. Beresnevich (University of York)

Diophantine approximation 18 January 2017 15 / 24

Admissible lattices and Bad

Let g be an (n + 1) × (n + 1) matrix such that Λ = gZn+1 is admissible. Let (x0, x1, . . . , xn) be any row (column) of g. Then x0 = 0 and x1 x0 , . . . , xn x0

• ∈ Bad(n, 1) .
• Proof. WLOG x0 = 1 and g is the first column.
• V. Beresnevich (University of York)

Diophantine approximation 18 January 2017 15 / 24

Admissible lattices and Bad

Let g be an (n + 1) × (n + 1) matrix such that Λ = gZn+1 is admissible. Let (x0, x1, . . . , xn) be any row (column) of g. Then x0 = 0 and x1 x0 , . . . , xn x0

• ∈ Bad(n, 1) .
• Proof. WLOG x0 = 1 and g is the first column. Let Q > 1 and

(q1, . . . , qn, p) ∈ Zn+1

=0 with |qi| ≤ Q.

• V. Beresnevich (University of York)

Diophantine approximation 18 January 2017 15 / 24

Admissible lattices and Bad

Let g be an (n + 1) × (n + 1) matrix such that Λ = gZn+1 is admissible. Let (x0, x1, . . . , xn) be any row (column) of g. Then x0 = 0 and x1 x0 , . . . , xn x0

• ∈ Bad(n, 1) .
• Proof. WLOG x0 = 1 and g is the first column. Let Q > 1 and

(q1, . . . , qn, p) ∈ Zn+1

=0 with |qi| ≤ Q. Recall,

0 < Nm(Λ) ≤ |λ0 · · · λn|, (λ1, . . . , λn) ∈ Λ=0 .

• V. Beresnevich (University of York)

Diophantine approximation 18 January 2017 15 / 24

Admissible lattices and Bad

Let g be an (n + 1) × (n + 1) matrix such that Λ = gZn+1 is admissible. Let (x0, x1, . . . , xn) be any row (column) of g. Then x0 = 0 and x1 x0 , . . . , xn x0

• ∈ Bad(n, 1) .
• Proof. WLOG x0 = 1 and g is the first column. Let Q > 1 and

(q1, . . . , qn, p) ∈ Zn+1

=0 with |qi| ≤ Q. Recall,

0 < Nm(Λ) ≤ |λ0 · · · λn|, (λ1, . . . , λn) ∈ Λ=0 . Let (λ0, . . . , λn)T = g(−p, q1, . . . , qn)T = 0.

• V. Beresnevich (University of York)

Diophantine approximation 18 January 2017 15 / 24

Admissible lattices and Bad

Let g be an (n + 1) × (n + 1) matrix such that Λ = gZn+1 is admissible. Let (x0, x1, . . . , xn) be any row (column) of g. Then x0 = 0 and x1 x0 , . . . , xn x0

• ∈ Bad(n, 1) .
• Proof. WLOG x0 = 1 and g is the first column. Let Q > 1 and

(q1, . . . , qn, p) ∈ Zn+1

=0 with |qi| ≤ Q. Recall,

0 < Nm(Λ) ≤ |λ0 · · · λn|, (λ1, . . . , λn) ∈ Λ=0 . Let (λ0, . . . , λn)T = g(−p, q1, . . . , qn)T = 0. Note |λi| ≤ CgQ.

• V. Beresnevich (University of York)

Diophantine approximation 18 January 2017 15 / 24

Admissible lattices and Bad

Let g be an (n + 1) × (n + 1) matrix such that Λ = gZn+1 is admissible. Let (x0, x1, . . . , xn) be any row (column) of g. Then x0 = 0 and x1 x0 , . . . , xn x0

• ∈ Bad(n, 1) .
• Proof. WLOG x0 = 1 and g is the first column. Let Q > 1 and

(q1, . . . , qn, p) ∈ Zn+1

=0 with |qi| ≤ Q. Recall,

0 < Nm(Λ) ≤ |λ0 · · · λn|, (λ1, . . . , λn) ∈ Λ=0 . Let (λ0, . . . , λn)T = g(−p, q1, . . . , qn)T = 0. Note |λi| ≤ CgQ. Then |λ0| =

• V. Beresnevich (University of York)

Diophantine approximation 18 January 2017 15 / 24

Admissible lattices and Bad

Let g be an (n + 1) × (n + 1) matrix such that Λ = gZn+1 is admissible. Let (x0, x1, . . . , xn) be any row (column) of g. Then x0 = 0 and x1 x0 , . . . , xn x0

• ∈ Bad(n, 1) .
• Proof. WLOG x0 = 1 and g is the first column. Let Q > 1 and

(q1, . . . , qn, p) ∈ Zn+1

=0 with |qi| ≤ Q. Recall,

0 < Nm(Λ) ≤ |λ0 · · · λn|, (λ1, . . . , λn) ∈ Λ=0 . Let (λ0, . . . , λn)T = g(−p, q1, . . . , qn)T = 0. Note |λi| ≤ CgQ. Then |λ0| = |q1x1 + · · · + qnxn − p| ≥

• V. Beresnevich (University of York)

Diophantine approximation 18 January 2017 15 / 24

Admissible lattices and Bad

Let g be an (n + 1) × (n + 1) matrix such that Λ = gZn+1 is admissible. Let (x0, x1, . . . , xn) be any row (column) of g. Then x0 = 0 and x1 x0 , . . . , xn x0

• ∈ Bad(n, 1) .
• Proof. WLOG x0 = 1 and g is the first column. Let Q > 1 and

(q1, . . . , qn, p) ∈ Zn+1

=0 with |qi| ≤ Q. Recall,

0 < Nm(Λ) ≤ |λ0 · · · λn|, (λ1, . . . , λn) ∈ Λ=0 . Let (λ0, . . . , λn)T = g(−p, q1, . . . , qn)T = 0. Note |λi| ≤ CgQ. Then |λ0| = |q1x1 + · · · + qnxn − p| ≥ |Nm(Λ)| |λ1 · · · λn| ≥

• V. Beresnevich (University of York)

Diophantine approximation 18 January 2017 15 / 24

Admissible lattices and Bad

Let g be an (n + 1) × (n + 1) matrix such that Λ = gZn+1 is admissible. Let (x0, x1, . . . , xn) be any row (column) of g. Then x0 = 0 and x1 x0 , . . . , xn x0

• ∈ Bad(n, 1) .
• Proof. WLOG x0 = 1 and g is the first column. Let Q > 1 and

(q1, . . . , qn, p) ∈ Zn+1

=0 with |qi| ≤ Q. Recall,

0 < Nm(Λ) ≤ |λ0 · · · λn|, (λ1, . . . , λn) ∈ Λ=0 . Let (λ0, . . . , λn)T = g(−p, q1, . . . , qn)T = 0. Note |λi| ≤ CgQ. Then |λ0| = |q1x1 + · · · + qnxn − p| ≥ |Nm(Λ)| |λ1 · · · λn| ≥ |Nm(Λ)| C n

g Qn

.

• V. Beresnevich (University of York)

Diophantine approximation 18 January 2017 15 / 24

Admissible lattices and Bad

Let g be an (n + 1) × (n + 1) matrix such that Λ = gZn+1 is admissible. Let (x0, x1, . . . , xn) be any row (column) of g. Then x0 = 0 and x1 x0 , . . . , xn x0

• ∈ Bad(n, 1) .
• Proof. WLOG x0 = 1 and g is the first column. Let Q > 1 and

(q1, . . . , qn, p) ∈ Zn+1

=0 with |qi| ≤ Q. Recall,

0 < Nm(Λ) ≤ |λ0 · · · λn|, (λ1, . . . , λn) ∈ Λ=0 . Let (λ0, . . . , λn)T = g(−p, q1, . . . , qn)T = 0. Note |λi| ≤ CgQ. Then |λ0| = |q1x1 + · · · + qnxn − p| ≥ |Nm(Λ)| |λ1 · · · λn| ≥ |Nm(Λ)| C n

g Qn

. Similarly one can give examples of matrices in Bad(n, m).

• V. Beresnevich (University of York)

Diophantine approximation 18 January 2017 15 / 24

Admissible lattices and Bad

Let g be an (n + 1) × (n + 1) matrix such that Λ = gZn+1 is admissible. Let (x0, x1, . . . , xn) be any row (column) of g. Then x0 = 0 and x1 x0 , . . . , xn x0

• ∈ Bad(n, 1) .
• Proof. WLOG x0 = 1 and g is the first column. Let Q > 1 and

(q1, . . . , qn, p) ∈ Zn+1

=0 with |qi| ≤ Q. Recall,

0 < Nm(Λ) ≤ |λ0 · · · λn|, (λ1, . . . , λn) ∈ Λ=0 . Let (λ0, . . . , λn)T = g(−p, q1, . . . , qn)T = 0. Note |λi| ≤ CgQ. Then |λ0| = |q1x1 + · · · + qnxn − p| ≥ |Nm(Λ)| |λ1 · · · λn| ≥ |Nm(Λ)| C n

g Qn

. Similarly one can give examples of matrices in Bad(n, m). The proof does not extend to weighted badly approximable points.

• V. Beresnevich (University of York)

Diophantine approximation 18 January 2017 15 / 24

More on Bad

Q: How likely is that a random matrix lies in Bad(n, m)? Answer:

• V. Beresnevich (University of York)

Diophantine approximation 18 January 2017 16 / 24

More on Bad

Q: How likely is that a random matrix lies in Bad(n, m)? Answer: Unlikely! The set Bad(n, m) has Lebesgue measure 0.

• V. Beresnevich (University of York)

Diophantine approximation 18 January 2017 16 / 24

More on Bad

Q: How likely is that a random matrix lies in Bad(n, m)? Answer: Unlikely! The set Bad(n, m) has Lebesgue measure 0. However: Bad(n, m) has Hausdorff dimension nm, the same as the dimension of the ambient space. (Schmidt, 1960s)

• V. Beresnevich (University of York)

Diophantine approximation 18 January 2017 16 / 24

More on Bad

Q: How likely is that a random matrix lies in Bad(n, m)? Answer: Unlikely! The set Bad(n, m) has Lebesgue measure 0. However: Bad(n, m) has Hausdorff dimension nm, the same as the dimension of the ambient space. (Schmidt, 1960s) Bad := Bad(1, 1) can be characterise using continued fractions:

• V. Beresnevich (University of York)

Diophantine approximation 18 January 2017 16 / 24

More on Bad

Q: How likely is that a random matrix lies in Bad(n, m)? Answer: Unlikely! The set Bad(n, m) has Lebesgue measure 0. However: Bad(n, m) has Hausdorff dimension nm, the same as the dimension of the ambient space. (Schmidt, 1960s) Bad := Bad(1, 1) can be characterise using continued fractions: x = a0 + 1 a1 + 1 a2 + ... = [a0; a1, a2, . . . ], where all an ∈ Z, an ≥ 1 for n = 1, 2, . . . .

• V. Beresnevich (University of York)

Diophantine approximation 18 January 2017 16 / 24

More on Bad

Q: How likely is that a random matrix lies in Bad(n, m)? Answer: Unlikely! The set Bad(n, m) has Lebesgue measure 0. However: Bad(n, m) has Hausdorff dimension nm, the same as the dimension of the ambient space. (Schmidt, 1960s) Bad := Bad(1, 1) can be characterise using continued fractions: x = a0 + 1 a1 + 1 a2 + ... = [a0; a1, a2, . . . ], where all an ∈ Z, an ≥ 1 for n = 1, 2, . . . . The convergents pn qn := [a1, a2, a3, . . . , an] are best approximations and satisfy 1 (an+1 + 2)q2

n

<

• x − pn

qn

• <

1 an+1q2

n

.

• V. Beresnevich (University of York)

Diophantine approximation 18 January 2017 16 / 24

More on Bad

Q: How likely is that a random matrix lies in Bad(n, m)? Answer: Unlikely! The set Bad(n, m) has Lebesgue measure 0. However: Bad(n, m) has Hausdorff dimension nm, the same as the dimension of the ambient space. (Schmidt, 1960s) Bad := Bad(1, 1) can be characterise using continued fractions: x = a0 + 1 a1 + 1 a2 + ... = [a0; a1, a2, . . . ], where all an ∈ Z, an ≥ 1 for n = 1, 2, . . . . The convergents pn qn := [a1, a2, a3, . . . , an] are best approximations and satisfy 1 (an+1 + 2)q2

n

<

• x − pn

qn

• <

1 an+1q2

n

. Fact: An irrational x = [a0; a1, a2 . . .] is in Bad ⇔ ∃ M ∀ i ∈ N ai ≤ M

• V. Beresnevich (University of York)

Diophantine approximation 18 January 2017 16 / 24

More on dimension 1

• V. Beresnevich (University of York)

Diophantine approximation 18 January 2017 17 / 24

More on dimension 1

Hurwitz’s Theorem (1891): ∀ x ∈ R \ Q there are infinitely many coprime integers p and q > 0 such that

• x − p

q

• <

1 √ 5q2 .

• V. Beresnevich (University of York)

Diophantine approximation 18 January 2017 17 / 24

More on dimension 1

Hurwitz’s Theorem (1891): ∀ x ∈ R \ Q there are infinitely many coprime integers p and q > 0 such that

• x − p

q

• <

1 √ 5q2 . The constant 1/( √ 5) is best possible. Recall: Bad := {x ∈ R : ∃ c(x) > 0 s.t.

• x − p

q

• ≥ c(x)

q2 ∀ q ∈ N, p ∈ Z} . quadratic irrationals are in Bad

• V. Beresnevich (University of York)

Diophantine approximation 18 January 2017 17 / 24

More on dimension 1

Hurwitz’s Theorem (1891): ∀ x ∈ R \ Q there are infinitely many coprime integers p and q > 0 such that

• x − p

q

• <

1 √ 5q2 . The constant 1/( √ 5) is best possible. Recall: Bad := {x ∈ R : ∃ c(x) > 0 s.t.

• x − p

q

• ≥ c(x)

q2 ∀ q ∈ N, p ∈ Z} . quadratic irrationals are in Bad Folklore Conjecture: Cubic irrationals are not in Bad

• V. Beresnevich (University of York)

Diophantine approximation 18 January 2017 17 / 24

More on dimension 1

Hurwitz’s Theorem (1891): ∀ x ∈ R \ Q there are infinitely many coprime integers p and q > 0 such that

• x − p

q

• <

1 √ 5q2 . The constant 1/( √ 5) is best possible. Recall: Bad := {x ∈ R : ∃ c(x) > 0 s.t.

• x − p

q

• ≥ c(x)

q2 ∀ q ∈ N, p ∈ Z} . quadratic irrationals are in Bad Folklore Conjecture: Cubic irrationals are not in Bad However, for any irrational algebraic x and any ε > 0 there exists a constant c(x, ε) > 0 such that (Roth, 1955)

• x − p

q

• ≥ c(x, ε)

q2+ε .

• V. Beresnevich (University of York)

Diophantine approximation 18 January 2017 17 / 24

More on dimension 1

Badly approximable numbers seem like worst approximable.

• V. Beresnevich (University of York)

Diophantine approximation 18 January 2017 18 / 24

More on dimension 1

Badly approximable numbers seem like worst approximable. However, they are the only irrationals that Dirichlet’s theorem can be improved for!!! Dirichlet’s Theorem (1848): Let x ∈ R \ Q and c = 1. Let Q > 1. Then there are integers p and q such that

• x − p

q

• < c

qQ , 1 ≤ q ≤ Q. (3)

• V. Beresnevich (University of York)

Diophantine approximation 18 January 2017 18 / 24

More on dimension 1

Badly approximable numbers seem like worst approximable. However, they are the only irrationals that Dirichlet’s theorem can be improved for!!! Dirichlet’s Theorem (1848): Let x ∈ R \ Q and c = 1. Let Q > 1. Then there are integers p and q such that

• x − p

q

• < c

qQ , 1 ≤ q ≤ Q. (3) Definition: x is called Dirichlet Improvable if there exists a positive c < 1 such that for all sufficiently large Q there are integers p and q satisfying (3).

• V. Beresnevich (University of York)

Diophantine approximation 18 January 2017 18 / 24

More on dimension 1

Badly approximable numbers seem like worst approximable. However, they are the only irrationals that Dirichlet’s theorem can be improved for!!! Dirichlet’s Theorem (1848): Let x ∈ R \ Q and c = 1. Let Q > 1. Then there are integers p and q such that

• x − p

q

• < c

qQ , 1 ≤ q ≤ Q. (3) Definition: x is called Dirichlet Improvable if there exists a positive c < 1 such that for all sufficiently large Q there are integers p and q satisfying (3). Theorem (Davenport and Schmidt (1970)): An irrational x is Dirichlet improvable if and only if x ∈ Bad.

• V. Beresnevich (University of York)

Diophantine approximation 18 January 2017 18 / 24

More on dimension 1

Badly approximable numbers seem like worst approximable. However, they are the only irrationals that Dirichlet’s theorem can be improved for!!! Dirichlet’s Theorem (1848): Let x ∈ R \ Q and c = 1. Let Q > 1. Then there are integers p and q such that

• x − p

q

• < c

qQ , 1 ≤ q ≤ Q. (3) Definition: x is called Dirichlet Improvable if there exists a positive c < 1 such that for all sufficiently large Q there are integers p and q satisfying (3). Theorem (Davenport and Schmidt (1970)): An irrational x is Dirichlet improvable if and only if x ∈ Bad.

• Remark. Dirichlet Improvable points/matrices can be introduced in higher

dimensions, but their characterisation is not that simple. It is known that almost every matrix is not Dirichlet Improvable.

• V. Beresnevich (University of York)

Diophantine approximation 18 January 2017 18 / 24

More on dimension 1: the Three Distance Theorem

To understand the gaps between qα − p, where α ∈ R \ Q is fixed and p, q ∈ Z, 0 ≤ q ≤ Q it is enough to describe the distribution of {α}, {2α}, . . . , {Qα} (4) where {·} denotes the fractional part.

• V. Beresnevich (University of York)

Diophantine approximation 18 January 2017 19 / 24

More on dimension 1: the Three Distance Theorem

To understand the gaps between qα − p, where α ∈ R \ Q is fixed and p, q ∈ Z, 0 ≤ q ≤ Q it is enough to describe the distribution of {α}, {2α}, . . . , {Qα} (4) where {·} denotes the fractional part. The Three Distance Theorem: For any α ∈ R \ Q and any integer Q ≥ 1 the points (4) partition [0, 1] into Q + 1 intervals which lengths take at most 3 different values δA, δB and δC with δC = δA + δB.

• V. Beresnevich (University of York)

Diophantine approximation 18 January 2017 19 / 24

More on dimension 1: the Three Distance Theorem

To understand the gaps between qα − p, where α ∈ R \ Q is fixed and p, q ∈ Z, 0 ≤ q ≤ Q it is enough to describe the distribution of {α}, {2α}, . . . , {Qα} (4) where {·} denotes the fractional part. The Three Distance Theorem: For any α ∈ R \ Q and any integer Q ≥ 1 the points (4) partition [0, 1] into Q + 1 intervals which lengths take at most 3 different values δA, δB and δC with δC = δA + δB. The length of the intervals, the number of intervals of each type and even the order in which the intervals of various type emerge can be determined using continued fractions!!!

• V. Beresnevich (University of York)

Diophantine approximation 18 January 2017 19 / 24

More on dimension 1: the Three Distance Theorem

Theorem: Let α ∈ R \ Q and [a0; a1, a2, . . . ] be the continued fraction expansion of α and pk/qk = [a0; a1, . . . , ak] and Dk = qkα − pk (k ≥ 0). Then for any Q ∈ N there exists a unique integer k ≥ 0 such that qk + qk−1 ≤ Q < qk+1 + qk (5) and unique integers r and s satisfying Q = rqk + qk−1 + s, 1 ≤ r ≤ ak+1 and 0 ≤ s ≤ qk − 1 (6) such that the points {α}, {2α}, . . . , {Qα} partition [0, 1] into Q + 1 intervals, of which NA = Q + 1 − qk are of length δA = |Dk|, NB = s + 1 are of length δB = |Dk+1| + (ak+1 − r)|Dk|, NC = qk − s − 1 are of length δC = δA + δB.

• V. Beresnevich (University of York)

Diophantine approximation 18 January 2017 20 / 24

More on dimension 1: Metric viewpoint

• V. Beresnevich (University of York)

Diophantine approximation 18 January 2017 21 / 24

More on dimension 1: Metric viewpoint

Let ψ : R+ → R+ and let W (ψ) be the set of x ∈ [0, 1] such that |qx − p| < ψ(q) (7) for infinitely many coprime (p, q) ∈ Z × N.

• V. Beresnevich (University of York)

Diophantine approximation 18 January 2017 21 / 24

More on dimension 1: Metric viewpoint

Let ψ : R+ → R+ and let W (ψ) be the set of x ∈ [0, 1] such that |qx − p| < ψ(q) (7) for infinitely many coprime (p, q) ∈ Z × N. Khintchine’s Theorem (1924) If ψ is monotonic, then m(W (ψ)) =    if ∞

q=1 ψ(q) < ∞ ,

1 if ∞

q=1 ψ(q) = ∞ .

• V. Beresnevich (University of York)

Diophantine approximation 18 January 2017 21 / 24

More on dimension 1: Metric viewpoint

Let ψ : R+ → R+ and let W (ψ) be the set of x ∈ [0, 1] such that |qx − p| < ψ(q) (7) for infinitely many coprime (p, q) ∈ Z × N. Khintchine’s Theorem (1924) If ψ is monotonic, then m(W (ψ)) =    if ∞

q=1 ψ(q) < ∞ ,

1 if ∞

q=1 ψ(q) = ∞ .

• Example. Take ψ(q) = q−1(log q)−1−ε with ε > 0.
• V. Beresnevich (University of York)

Diophantine approximation 18 January 2017 21 / 24

More on dimension 1: Metric viewpoint

Let ψ : R+ → R+ and let W (ψ) be the set of x ∈ [0, 1] such that |qx − p| < ψ(q) (7) for infinitely many coprime (p, q) ∈ Z × N. Khintchine’s Theorem (1924) If ψ is monotonic, then m(W (ψ)) =    if ∞

q=1 ψ(q) < ∞ ,

1 if ∞

q=1 ψ(q) = ∞ .

• Example. Take ψ(q) = q−1(log q)−1−ε with ε > 0. The Khintchine sum
• converges. Hence for x picked at random (7) has only a finite number of
• solution. Also means that for almost every x there exists a constant

c(x, ψ) > 0 such that |qx − p| ≥ c(x, ψ) (q log q)1+ε for all (p, q) ∈ Z × N.

• V. Beresnevich (University of York)

Diophantine approximation 18 January 2017 21 / 24

The Khintchine-Groshev theorem

• V. Beresnevich (University of York)

Diophantine approximation 18 January 2017 22 / 24

The Khintchine-Groshev theorem

Let ψ : R>0 → R>0 and W (n, m; ψ) denote the set of real matrices X = (xi,j)1≤j≤n

≤i≤m

with 0 ≤ xi,j ≤ 1 such that |q1αi,1 + · · · + qnαi,n − pi| < ψ(Q) (1 ≤ i ≤ m) 1 ≤ max

1≤j≤n |qj| ≤ Q

holds for infinitely many (q1, . . . , qn, p1, . . . , pm) ∈ Zn+m.

• V. Beresnevich (University of York)

Diophantine approximation 18 January 2017 22 / 24

The Khintchine-Groshev theorem

Let ψ : R>0 → R>0 and W (n, m; ψ) denote the set of real matrices X = (xi,j)1≤j≤n

≤i≤m

with 0 ≤ xi,j ≤ 1 such that |q1αi,1 + · · · + qnαi,n − pi| < ψ(Q) (1 ≤ i ≤ m) 1 ≤ max

1≤j≤n |qj| ≤ Q

holds for infinitely many (q1, . . . , qn, p1, . . . , pm) ∈ Zn+m. Recall that Dirichlet’s theorem ⇒ W (n, m)(ψ) is everything if ψ(Q) = Q−n/m.

• V. Beresnevich (University of York)

Diophantine approximation 18 January 2017 22 / 24

The Khintchine-Groshev theorem

Let ψ : R>0 → R>0 and W (n, m; ψ) denote the set of real matrices X = (xi,j)1≤j≤n

≤i≤m

with 0 ≤ xi,j ≤ 1 such that |q1αi,1 + · · · + qnαi,n − pi| < ψ(Q) (1 ≤ i ≤ m) 1 ≤ max

1≤j≤n |qj| ≤ Q

holds for infinitely many (q1, . . . , qn, p1, . . . , pm) ∈ Zn+m. Recall that Dirichlet’s theorem ⇒ W (n, m)(ψ) is everything if ψ(Q) = Q−n/m. Theorem (Khintchine-Groshev /1924-1938/): Suppose that ψ is

• monotonic. Then

Prob(X ∈ W (n, m; ψ)) =    0, if ∞

q=1 qn−1ψ(q)m < ∞ ,

1, if ∞

q=1 qn−1ψ(q)m = ∞ .

• V. Beresnevich (University of York)

Diophantine approximation 18 January 2017 22 / 24

The Khintchine-Groshev theorem for manifolds

• V. Beresnevich (University of York)

Diophantine approximation 18 January 2017 23 / 24

The Khintchine-Groshev theorem for manifolds

Q: What can be said about the probability of W (n, m; ψ) under the condition that the entries of the matrix X are functionally related?

• V. Beresnevich (University of York)

Diophantine approximation 18 January 2017 23 / 24

The Khintchine-Groshev theorem for manifolds

Q: What can be said about the probability of W (n, m; ψ) under the condition that the entries of the matrix X are functionally related? Functional relations ⇒ that X lies on a submanifold (curve or surface).

• V. Beresnevich (University of York)

Diophantine approximation 18 January 2017 23 / 24

The Khintchine-Groshev theorem for manifolds

Q: What can be said about the probability of W (n, m; ψ) under the condition that the entries of the matrix X are functionally related? Functional relations ⇒ that X lies on a submanifold (curve or surface). We’ll restrict to the case of one linear form: m = 1.

• V. Beresnevich (University of York)

Diophantine approximation 18 January 2017 23 / 24

The Khintchine-Groshev theorem for manifolds

Q: What can be said about the probability of W (n, m; ψ) under the condition that the entries of the matrix X are functionally related? Functional relations ⇒ that X lies on a submanifold (curve or surface). We’ll restrict to the case of one linear form: m = 1. Also restrict ourselves to the case when the manifold is analytic, e.g. is given by polynomial relations. The coordinates of X = (x1, . . . , xn) are analytic functions of several (possibly just one) other variables. Example: X = (x, x2, x3, . . . , xn).

• V. Beresnevich (University of York)

Diophantine approximation 18 January 2017 23 / 24

The Khintchine-Groshev theorem for manifolds

Q: What can be said about the probability of W (n, m; ψ) under the condition that the entries of the matrix X are functionally related? Functional relations ⇒ that X lies on a submanifold (curve or surface). We’ll restrict to the case of one linear form: m = 1. Also restrict ourselves to the case when the manifold is analytic, e.g. is given by polynomial relations. The coordinates of X = (x1, . . . , xn) are analytic functions of several (possibly just one) other variables. Example: X = (x, x2, x3, . . . , xn). Non-degeneracy: A connected analytic submanifold M of Rn is non-degenerate if it is not entirely contained in a hyperplane of Rn.

• V. Beresnevich (University of York)

Diophantine approximation 18 January 2017 23 / 24

The Khintchine-Groshev theorem for manifolds

Q: What can be said about the probability of W (n, m; ψ) under the condition that the entries of the matrix X are functionally related? Functional relations ⇒ that X lies on a submanifold (curve or surface). We’ll restrict to the case of one linear form: m = 1. Also restrict ourselves to the case when the manifold is analytic, e.g. is given by polynomial relations. The coordinates of X = (x1, . . . , xn) are analytic functions of several (possibly just one) other variables. Example: X = (x, x2, x3, . . . , xn). Non-degeneracy: A connected analytic submanifold M of Rn is non-degenerate if it is not entirely contained in a hyperplane of Rn. The Khintchine-Groshev theorem for non-degenerate manifolds (BBKM, 1998-2002): Suppose that ψ is monotonic and M ⊂ Rn is non-degenerate. Then Prob(X ∈ W (n, 1; ψ)|X ∈ M) =    0, if ∞

q=1 qn−1ψ(q) < ∞ ,

1, if ∞

q=1 qn−1ψ(q) = ∞ .

• V. Beresnevich (University of York)

Diophantine approximation 18 January 2017 23 / 24

Extremal manifolds and homogeneous flows

• V. Beresnevich (University of York)

Diophantine approximation 18 January 2017 24 / 24

Extremal manifolds and homogeneous flows

Consider ψτ(q) = q−τ

• V. Beresnevich (University of York)

Diophantine approximation 18 January 2017 24 / 24

Extremal manifolds and homogeneous flows

Consider ψτ(q) = q−τ Extremal matrices: X is extremal if X ∈ W (n, m; ψτ) for any τ > n/m. If X is not extremal it is called very well approximable (VWA).

• V. Beresnevich (University of York)

Diophantine approximation 18 January 2017 24 / 24

Extremal manifolds and homogeneous flows

Consider ψτ(q) = q−τ Extremal matrices: X is extremal if X ∈ W (n, m; ψτ) for any τ > n/m. If X is not extremal it is called very well approximable (VWA). Theorem of Kleinbock and Margulis, 1998): Almost every point on any non-degenerate manifold M in Rn is extremal.

• V. Beresnevich (University of York)

Diophantine approximation 18 January 2017 24 / 24

Extremal manifolds and homogeneous flows

Consider ψτ(q) = q−τ Extremal matrices: X is extremal if X ∈ W (n, m; ψτ) for any τ > n/m. If X is not extremal it is called very well approximable (VWA). Theorem of Kleinbock and Margulis, 1998): Almost every point on any non-degenerate manifold M in Rn is extremal. Relation to homogeneous flows: Let gt = diag{et, e−t/n, . . . , e−t/n} and ΛX =      1 x1 . . . xn 1 . . . ... . . . 1      Zn+1

• V. Beresnevich (University of York)

Diophantine approximation 18 January 2017 24 / 24

Extremal manifolds and homogeneous flows

Consider ψτ(q) = q−τ Extremal matrices: X is extremal if X ∈ W (n, m; ψτ) for any τ > n/m. If X is not extremal it is called very well approximable (VWA). Theorem of Kleinbock and Margulis, 1998): Almost every point on any non-degenerate manifold M in Rn is extremal. Relation to homogeneous flows: Let gt = diag{et, e−t/n, . . . , e−t/n} and ΛX =      1 x1 . . . xn 1 . . . ... . . . 1      Zn+1 Let δ(Λ) = inf{λ : λ ∈ Λ=0} .

• V. Beresnevich (University of York)

Diophantine approximation 18 January 2017 24 / 24

Extremal manifolds and homogeneous flows

Consider ψτ(q) = q−τ Extremal matrices: X is extremal if X ∈ W (n, m; ψτ) for any τ > n/m. If X is not extremal it is called very well approximable (VWA). Theorem of Kleinbock and Margulis, 1998): Almost every point on any non-degenerate manifold M in Rn is extremal. Relation to homogeneous flows: Let gt = diag{et, e−t/n, . . . , e−t/n} and ΛX =      1 x1 . . . xn 1 . . . ... . . . 1      Zn+1 Let δ(Λ) = inf{λ : λ ∈ Λ=0} . Fact: X is VWA if and only if there exists ε > such that δ(gtΛX) ≤ e−εt for arbitrarily large t.

• V. Beresnevich (University of York)

Diophantine approximation 18 January 2017 24 / 24