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◆✉♠❜❡r t❤❡♦r② ❛♥❞ s♣❡❝tr❛❧ ✐♥st❛❜✐❧✐t✐❡s ✐♥ ♠❡t❛♠❛t❡r✐❛❧s

Štˇ epán Starosta

Faculty of Information Technology Czech Technical University in Prague

AAMP13 2016, Prague

1 / 16

Motivation Markov constant Results

❚❤❡ s❡t S(α)

When analyzing the spectrum of the operator

f(x, y) = λf(x, y) , = ∂2 ∂x2 − ∂2 ∂y2 ,

f|∂R = 0, with R being a rectangle with sides a and b: b a R

2 / 16

Motivation Markov constant Results

❚❤❡ s❡t S(α)

When analyzing the spectrum of the operator

f(x, y) = λf(x, y) , = ∂2 ∂x2 − ∂2 ∂y2 ,

f|∂R = 0, with R being a rectangle with sides a and b: b a R the following set comes up

S(α) = set of all accumulation points of

• m2
• k

m − α

• : k, m ∈ Z
• ,

where α = a

b ∈ R.

[Talk of D. Krejˇ ciˇ rík]

2 / 16

Motivation Markov constant Results

❈♦♥t✐♥✉❡❞ ❢r❛❝t✐♦♥s

For α ∈ R, the continued fraction expansion of α is the sequence

(ai)i=0 with ai ∈ Z, aj > 0 for j > 0 such that α = a0 +

1 a1 + 1 a2 + 1 a3 + · · ·

.

Notation: α = [a0, a1, a2, . . .].

3 / 16

Motivation Markov constant Results

❈♦♥t✐♥✉❡❞ ❢r❛❝t✐♦♥s

For α ∈ R, the continued fraction expansion of α is the sequence

(ai)i=0 with ai ∈ Z, aj > 0 for j > 0 such that α = a0 +

1 a1 + 1 a2 + 1 a3 + · · ·

.

Notation: α = [a0, a1, a2, . . .]. ai is called a partial coefficient.

3 / 16

Motivation Markov constant Results

❈♦♥t✐♥✉❡❞ ❢r❛❝t✐♦♥s

For α ∈ R, the continued fraction expansion of α is the sequence

(ai)i=0 with ai ∈ Z, aj > 0 for j > 0 such that α = a0 +

1 a1 + 1 a2 + 1 a3 + · · ·

.

Notation: α = [a0, a1, a2, . . .]. ai is called a partial coefficient. The N-th convergent of α is pN qN

= a0 +

1 a1 + 1 a2 + 1 ... + 1 aN

∈ Q.

3 / 16

Motivation Markov constant Results

▼❛r❦♦✈ ❝♦♥st❛♥t

S(α) = set of all accumulation points of

• m2 k

m − α

• : k, m ∈ Z
• The Markov constant of α is

µ(α) = lim inf

m→+∞ m mα =

if α ∈ Q min |S(α)| , where x is the distance to the nearest integer

x = min{|x − n|: n ∈ Z}

4 / 16

Motivation Markov constant Results

▼❛r❦♦✈ ❝♦♥st❛♥t

S(α) = set of all accumulation points of

• m2 k

m − α

• : k, m ∈ Z
• The Markov constant of α is

µ(α) = lim inf

m→+∞ m mα =

if α ∈ Q min |S(α)| , where x is the distance to the nearest integer

x = min{|x − n|: n ∈ Z}

L = {µ(α): α ∈ R} (L is sometimes called the Lagrange spectrum)

4 / 16

Motivation Markov constant Results

❇❛s✐❝ r❡s✉❧ts ♦♥ t❤❡ s❡t L

Hurwitz’s theorem: there exists infinitely many p, q ∈ Z such that

• α − p

q

• <

1

√

5q2 (also follows from an earlier result of Markov)

5 / 16

Motivation Markov constant Results

❇❛s✐❝ r❡s✉❧ts ♦♥ t❤❡ s❡t L

Hurwitz’s theorem: there exists infinitely many p, q ∈ Z such that

• α − p

q

• <

1

√

5q2 (also follows from an earlier result of Markov) thus, µ(α) ≤

1

√

5

5 / 16

Motivation Markov constant Results

❇❛s✐❝ r❡s✉❧ts ♦♥ t❤❡ s❡t L

Hurwitz’s theorem: there exists infinitely many p, q ∈ Z such that

• α − p

q

• <

1

√

5q2 (also follows from an earlier result of Markov) thus, µ(α) ≤

1

√

5

if α is not equivalent to 1+

√

5 2

= [1, 1, 1, . . .], then we may replace √

5 by

√

8

5 / 16

Motivation Markov constant Results

❇❛s✐❝ r❡s✉❧ts ♦♥ t❤❡ s❡t L

Hurwitz’s theorem: there exists infinitely many p, q ∈ Z such that

• α − p

q

• <

1

√

5q2 (also follows from an earlier result of Markov) thus, µ(α) ≤

1

√

5

if α is not equivalent to 1+

√

5 2

= [1, 1, 1, . . .], then we may replace √

5 by

√

8 if, moreover, α is not equivalent to 1 +

√

2 = [2, 2, 2, . . .], then we may replace

√

8 by

√

221 5

5 / 16

Motivation Markov constant Results

❈❛❧❝✉❧❛t✐♦♥ ♦❢ µ(α)

Let pN

qN be the convergents of α = [a0, a1, a2, . . .], then

q2

N

• α − pN

qN

• = (−1)N

1

[aN+1, aN+2, . . .] + [0, aN, . . . , a1].

6 / 16

Motivation Markov constant Results

❈❛❧❝✉❧❛t✐♦♥ ♦❢ µ(α)

Let pN

qN be the convergents of α = [a0, a1, a2, . . .], then

q2

N

• α − pN

qN

• = (−1)N

1

[aN+1, aN+2, . . .] + [0, aN, . . . , a1]. µ(α) = µ(α) = lim infm→+∞ m mα

lim inf

N→+∞

1

[aN+1, aN+2, . . .] + [0, aN, . . . , a1]

6 / 16

Motivation Markov constant Results

❈❛❧❝✉❧❛t✐♦♥ ♦❢ µ(α)

Let pN

qN be the convergents of α = [a0, a1, a2, . . .], then

q2

N

• α − pN

qN

• = (−1)N

1

[aN+1, aN+2, . . .] + [0, aN, . . . , a1]. µ(α) = µ(α) = lim infm→+∞ m mα

lim inf

N→+∞

1

[aN+1, aN+2, . . .] + [0, aN, . . . , a1] µ

• 1 +

√

5 2

• =

1

1+

√

5 2

+ 1+

√

5 2

− 1 =

1

√

5

6 / 16

Motivation Markov constant Results

❈❛❧❝✉❧❛t✐♦♥ ♦❢ µ(α)

Let pN

qN be the convergents of α = [a0, a1, a2, . . .], then

q2

N

• α − pN

qN

• = (−1)N

1

[aN+1, aN+2, . . .] + [0, aN, . . . , a1]. µ(α) = µ(α) = lim infm→+∞ m mα

lim inf

N→+∞

1

[aN+1, aN+2, . . .] + [0, aN, . . . , a1] µ

• 1 +

√

5 2

• =

1

1+

√

5 2

+ 1+

√

5 2

− 1 =

1

√

5

µ

• 1 +

√

2

• =

1 1 +

√

2 + 1 +

√

2 − 2 = 1 2

√

2 = 1

√

8

6 / 16

Motivation Markov constant Results

❚❤❡ s❡t L

7 / 16

Motivation Markov constant Results

❚❤❡ s❡t L

1

√

5

7 / 16

Motivation Markov constant Results

❚❤❡ s❡t L

1

√

5

√

1

√

8

7 / 16

Motivation Markov constant Results

❚❤❡ s❡t L

1

√

5

√

√

1

√

8 5

√

221

7 / 16

Motivation Markov constant Results

❚❤❡ s❡t L

1

√

5

√

√

√

1

√

8 5

√

221 13

√

1571

7 / 16

Motivation Markov constant Results

❚❤❡ s❡t L

1

√

5

√

√

√

1 3

largest accumulation point

1

√

8 5

√

221 13

√

1571

1

7 / 16

Motivation Markov constant Results

❚❤❡ s❡t L

1

√

5

√

√

√

1 3

largest accumulation point discrete part 7 / 16

Motivation Markov constant Results

❚❤❡ s❡t L

1

√

5

√

√

√

1 3

largest accumulation point discrete part

1 √ 12

partial coefficients in {1, 2} 7 / 16

Motivation Markov constant Results

❚❤❡ s❡t L

1

√

5

√

√

√

1 3

largest accumulation point discrete part

1 √ 12

partial coefficients in {1, 2}

1 √ 13

at least one partial coefficient larger than 2 7 / 16

Motivation Markov constant Results

❚❤❡ s❡t L

1

√

5

√

√

√

1 3

largest accumulation point discrete part

1 √ 12

partial coefficients in {1, 2}

1 √ 13

at least one partial coefficient larger than 2

F F =

555391024−70937

√

462 2507812168

≈ 0.22

(Freiman’s constant)

7 / 16

Motivation Markov constant Results

❚❤❡ s❡t L

1

√

5

√

√

√

1 3

largest accumulation point discrete part

1 √ 12

partial coefficients in {1, 2}

1 √ 13

at least one partial coefficient larger than 2

F F =

555391024−70937

√

462 2507812168

≈ 0.22

(Freiman’s constant)

continuous part (Hall’s ray) 7 / 16

Motivation Markov constant Results

❚❤❡ s❡t L

1

√

5

√

√

√

1 3

largest accumulation point discrete part

1 √ 12

partial coefficients in {1, 2}

1 √ 13

at least one partial coefficient larger than 2

F F =

555391024−70937

√

462 2507812168

≈ 0.22

(Freiman’s constant)

continuous part (Hall’s ray) explored jungle largest gap 7 / 16

Motivation Markov constant Results

❱❛r✐♦✉s ❜❡❤❛✈✐♦✉r ♦❢ S(α)

✭❥♦✐♥t ✇♦r❦ ✇✐t❤ ❊✳ P❡❧❛♥t♦✈á ❛♥❞ ▼✳ ❩♥♦❥✐❧✮ S(α) = set of all accumulation points of

• m2 k

m − α

• : k, m ∈ Z
• S(α) is a closed set, closed under multiplication by squares

8 / 16

Motivation Markov constant Results

❱❛r✐♦✉s ❜❡❤❛✈✐♦✉r ♦❢ S(α)

✭❥♦✐♥t ✇♦r❦ ✇✐t❤ ❊✳ P❡❧❛♥t♦✈á ❛♥❞ ▼✳ ❩♥♦❥✐❧✮ S(α) = set of all accumulation points of

• m2 k

m − α

• : k, m ∈ Z
• S(α) is a closed set, closed under multiplication by squares
• α ∈ Q ⇒ S(α) = ∅

8 / 16

Motivation Markov constant Results

❱❛r✐♦✉s ❜❡❤❛✈✐♦✉r ♦❢ S(α)

✭❥♦✐♥t ✇♦r❦ ✇✐t❤ ❊✳ P❡❧❛♥t♦✈á ❛♥❞ ▼✳ ❩♥♦❥✐❧✮ S(α) = set of all accumulation points of

• m2 k

m − α

• : k, m ∈ Z
• S(α) is a closed set, closed under multiplication by squares
• α ∈ Q ⇒ S(α) = ∅
• α /

∈ Q ⇒ min |S(α)| = µ(α) and S(α) is not empty

8 / 16

Motivation Markov constant Results

❱❛r✐♦✉s ❜❡❤❛✈✐♦✉r ♦❢ S(α)

✭❥♦✐♥t ✇♦r❦ ✇✐t❤ ❊✳ P❡❧❛♥t♦✈á ❛♥❞ ▼✳ ❩♥♦❥✐❧✮ S(α) = set of all accumulation points of

• m2 k

m − α

• : k, m ∈ Z
• S(α) is a closed set, closed under multiplication by squares
• α ∈ Q ⇒ S(α) = ∅
• α /

∈ Q ⇒ min |S(α)| = µ(α) and S(α) is not empty

• If a, b, c, d ∈ Z such that ad − bc = ∆ = 0, then

∆ · S(α) ⊂ S

• aα + b

cα + d

• .

8 / 16

Motivation Markov constant Results

❱❛r✐♦✉s ❜❡❤❛✈✐♦✉r ♦❢ S(α)

✭❥♦✐♥t ✇♦r❦ ✇✐t❤ ❊✳ P❡❧❛♥t♦✈á ❛♥❞ ▼✳ ❩♥♦❥✐❧✮ S(α) = set of all accumulation points of

• m2 k

m − α

• : k, m ∈ Z
• S(α) is a closed set, closed under multiplication by squares
• α ∈ Q ⇒ S(α) = ∅
• α /

∈ Q ⇒ min |S(α)| = µ(α) and S(α) is not empty

• If a, b, c, d ∈ Z such that ad − bc = ∆ = 0, then

∆ · S(α) ⊂ S

• aα + b

cα + d

• .
• S([an+k, an+1+k, an+2+k, . . .]) =

(−1)kS([an, an+1, an+2, . . .]) for any k, n ∈ N

8 / 16

Motivation Markov constant Results

❱❛r✐♦✉s ❜❡❤❛✈✐♦✉r ♦❢ S(α)

✭❥♦✐♥t ✇♦r❦ ✇✐t❤ ❊✳ P❡❧❛♥t♦✈á ❛♥❞ ▼✳ ❩♥♦❥✐❧✮ S(α) = set of all accumulation points of

• m2 k

m − α

• : k, m ∈ Z
• S(α) is a closed set, closed under multiplication by squares
• α ∈ Q ⇒ S(α) = ∅
• α /

∈ Q ⇒ min |S(α)| = µ(α) and S(α) is not empty

• If a, b, c, d ∈ Z such that ad − bc = ∆ = 0, then

∆ · S(α) ⊂ S

• aα + b

cα + d

• .
• S([an+k, an+1+k, an+2+k, . . .]) =

(−1)kS([an, an+1, an+2, . . .]) for any k, n ∈ N

• If α /

∈ Q and I is an interval, then there exists β ∈ I such that S(α) = S(β).

8 / 16

Motivation Markov constant Results

S(α) ❛♥❞ ❝♦♥t✐♥✉❡❞ ❢r❛❝t✐♦♥ ❡①♣❛♥s✐♦♥ ♦❢ α

points of S(α) ∩

• − 1

2, 1 2

• are accumulation points of

q2

N

• α − pN

qN

• = (−1)N

1

[aN+1, aN+2, . . .] + [0, aN, . . . , a1].

where pN

qN are the convergents of α = [a0, a1, a2, . . .]

9 / 16

Motivation Markov constant Results

❲❡❧❧✲❛♣♣r♦①✐♠❛❜❧❡ ♥✉♠❜❡rs

α is well approximable it it has unbounded partial coefficients ❡ ❡ ❡

10 / 16

Motivation Markov constant Results

❲❡❧❧✲❛♣♣r♦①✐♠❛❜❧❡ ♥✉♠❜❡rs

α is well approximable it it has unbounded partial coefficients α is well approximable

α/ ∈Q

⇔ µ(α) = 0 ❡ ❡ ❡

10 / 16

Motivation Markov constant Results

❲❡❧❧✲❛♣♣r♦①✐♠❛❜❧❡ ♥✉♠❜❡rs

α is well approximable it it has unbounded partial coefficients α is well approximable

α/ ∈Q

⇔ µ(α) = 0 ⇔ 0 ∈ S(α) ❡ ❡ ❡

10 / 16

Motivation Markov constant Results

❲❡❧❧✲❛♣♣r♦①✐♠❛❜❧❡ ♥✉♠❜❡rs

α is well approximable it it has unbounded partial coefficients α is well approximable

α/ ∈Q

⇔ µ(α) = 0 ⇔ 0 ∈ S(α)

Example: for ❡ = [2, 1, 2, 1, 1, 4, 1, 1, 6, 1, 1, 8, 1, 1, 10, . . .] we have

(− 1

2, 1 2) ∩ S(❡) = {0}

and

{a + 1

2 : a ∈ Z} ⊂ S(❡).

10 / 16

Motivation Markov constant Results

❲❡❧❧✲❛♣♣r♦①✐♠❛❜❧❡ ♥✉♠❜❡rs

α is well approximable it it has unbounded partial coefficients α is well approximable

α/ ∈Q

⇔ µ(α) = 0 ⇔ 0 ∈ S(α)

Example: for ❡ = [2, 1, 2, 1, 1, 4, 1, 1, 6, 1, 1, 8, 1, 1, 10, . . .] we have

(− 1

2, 1 2) ∩ S(❡) = {0}

and

{a + 1

2 : a ∈ Z} ⊂ S(❡).

Let α be an irrational well approximable number. For any n ∈ N the interval [n, n + 1] or the interval [−n − 1, −n] has a non-empty intersection with S(α).

10 / 16

Motivation Markov constant Results

❇❛❞❧② ❛♣♣r♦①✐♠❛❜❧❡ ♥✉♠❜❡rs

If an irrational number is not well approximable, it is badly approximable.

11 / 16

Motivation Markov constant Results

❇❛❞❧② ❛♣♣r♦①✐♠❛❜❧❡ ♥✉♠❜❡rs

If an irrational number is not well approximable, it is badly approximable. Recall

S(α)∩

• −1

2, 1 2

• = acc. pts of
• (−1)N

[aN+1, aN+2, . . .] + [0, aN, . . . , a1]

• 11 / 16

Motivation Markov constant Results

❇❛❞❧② ❛♣♣r♦①✐♠❛❜❧❡ ♥✉♠❜❡rs

If an irrational number is not well approximable, it is badly approximable. Recall

S(α)∩

• −1

2, 1 2

• = acc. pts of
• (−1)N

[aN+1, aN+2, . . .] + [0, aN, . . . , a1]

• Set F(m) = {[t, a1, a2, a3, . . .]: t ∈ Z, 1 ≤ ai ≤ m }

11 / 16

Motivation Markov constant Results

❈❛♥t♦r s❡t

1 0 •3 1 . . . 0 •3 2 . . .

12 / 16

Motivation Markov constant Results

❈❛♥t♦r s❡t

1 0 •3 1 . . . 0 •3 2 . . . 0 •3 11 . . . 0 •3 12 . . . 0 •3 21 . . . 0 •3 22 . . .

12 / 16

Motivation Markov constant Results

❈❛♥t♦r s❡t

1 0 •3 1 . . . 0 •3 2 . . . 0 •3 11 . . . 0 •3 12 . . . 0 •3 21 . . . 0 •3 22 . . . 0 •3 111 . . . 0 •3 112 . . . 0 •3 121 . . . 0 •3 122 . . . 0 •3 211 . . . 0 •3 212 . . . 0 •3 221 . . . 0 •3 222 . . .

12 / 16

Motivation Markov constant Results

❚❤❡ s❡t F(2)

F(2) = {[t, a1, a2, a3, . . .]: t ∈ Z, ai ∈ {1, 2} } 1

[0, 1, . . .] [0, 2, . . .]

13 / 16

Motivation Markov constant Results

❚❤❡ s❡t F(2)

F(2) = {[t, a1, a2, a3, . . .]: t ∈ Z, ai ∈ {1, 2} } 1

[0, 1, . . .] [0, 2, . . .] [0, 1, 1, . . .] [0, 1, 2, . . .] [0, 2, 1, . . .] [0, 2, 2, . . .]

13 / 16

Motivation Markov constant Results

❚❤❡ s❡t F(2)

F(2) = {[t, a1, a2, a3, . . .]: t ∈ Z, ai ∈ {1, 2} } 1

[0, 1, . . .] [0, 2, . . .] [0, 1, 1, . . .] [0, 1, 2, . . .] [0, 2, 1, . . .] [0, 2, 2, . . .] [0, 1, 1, 2 . . .] [0, 1, 1, 1 . . .] [0, 1, 2, 2 . . .] [0, 1, 2, 1 . . .] [0, 2, 1, 2 . . .] [0, 2, 1, 1 . . .] [0, 2, 2, 2 . . .] [0, 2, 2, 1 . . .]

13 / 16

Motivation Markov constant Results

❑♥♦✇ r❡s✉❧ts ♦♥ F(m)

F(m) = {[t, a1, a2, a3, . . .]: t ∈ Z, 1 ≤ ai ≤ m }

14 / 16

Motivation Markov constant Results

❑♥♦✇ r❡s✉❧ts ♦♥ F(m)

F(m) = {[t, a1, a2, a3, . . .]: t ∈ Z, 1 ≤ ai ≤ m } F(4) + F(4) = R

14 / 16

Motivation Markov constant Results

❑♥♦✇ r❡s✉❧ts ♦♥ F(m)

F(m) = {[t, a1, a2, a3, . . .]: t ∈ Z, 1 ≤ ai ≤ m } F(4) + F(4) = R F(3) + F(4) = R = F(2) + F(5)

14 / 16

Motivation Markov constant Results

❑♥♦✇ r❡s✉❧ts ♦♥ F(m)

F(m) = {[t, a1, a2, a3, . . .]: t ∈ Z, 1 ≤ ai ≤ m } F(4) + F(4) = R F(3) + F(4) = R = F(2) + F(5) F(3) + F(3) = R = F(2) + F(4)

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Motivation Markov constant Results

❇❛❞❧② ❛♣♣r♦①✐♠❛❜❧❡ ♥✉♠❜❡rs ■■

For each property in the following list there exists a badly approximable irrational number α satisfying the property:

1 S(α) = R; 2 S(α) = (−∞, −ε] ∪ [ε, +∞), where ε =

√

2 8 ≈ 0.18; 3 the Hausdorff dimension of S(α) ∩

• − 1

2, 1 2

• is positive but less

than 1.

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Motivation Markov constant Results

❇❛❞❧② ❛♣♣r♦①✐♠❛❜❧❡ ♥✉♠❜❡rs ■■

For each property in the following list there exists a badly approximable irrational number α satisfying the property:

1 S(α) = R; 2 S(α) = (−∞, −ε] ∪ [ε, +∞), where ε =

√

2 8 ≈ 0.18; 3 the Hausdorff dimension of S(α) ∩

• − 1

2, 1 2

• is positive but less

than 1. For almost all α ∈ R, S(α) = R.

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Motivation Markov constant Results

α q✉❛❞r❛t✐❝

✭❥♦✐♥t ✇♦r❦ ✇✐t❤ ❱✳ ❑á❧❛✱ ✇♦r❦ ✐♥ ♣r♦❣r❡ss✮ Assume α is quadratic: the continued fraction expansion of α is ultimately periodic.

◆

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Motivation Markov constant Results

α q✉❛❞r❛t✐❝

✭❥♦✐♥t ✇♦r❦ ✇✐t❤ ❱✳ ❑á❧❛✱ ✇♦r❦ ✐♥ ♣r♦❣r❡ss✮ Assume α is quadratic: the continued fraction expansion of α is ultimately periodic. This fact may be used to explicitly describe the set S(α) ∩

• − 1

2, 1 2

• .

◆

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Motivation Markov constant Results

α q✉❛❞r❛t✐❝

✭❥♦✐♥t ✇♦r❦ ✇✐t❤ ❱✳ ❑á❧❛✱ ✇♦r❦ ✐♥ ♣r♦❣r❡ss✮ Assume α is quadratic: the continued fraction expansion of α is ultimately periodic. This fact may be used to explicitly describe the set S(α) ∩

• − 1

2, 1 2

• .

A different approach leads to the following

S(α) = S(α′) = C ·

• ◆Q(α)/Q(k + mα): k, m ∈ Z
• for some constant C depending on α.

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