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Uniform distribution: approximating continuous objects by discrete ones Dmitriy Bilyk School of Mathematics, University of Minnesota Minneapolis, MN Introduction to Research seminar February 10, 2016 Dmitriy Bilyk Uniform distribution:

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SLIDE 1

Uniform distribution: approximating continuous objects by discrete ones

Dmitriy Bilyk School of Mathematics, University of Minnesota Minneapolis, MN “Introduction to Research” seminar February 10, 2016

Dmitriy Bilyk Uniform distribution: discrete vs. continuous

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SLIDE 2

Uniform distribution of sequences

A sequence (xn) is uniformly distributed in [0, 1] iff for any interval I ⊂ [0, 1] : lim

N→∞

#{n ≤ N : xn ∈ I} N = |I|

Dmitriy Bilyk Uniform distribution: discrete vs. continuous

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SLIDE 3

Uniform distribution of sequences

A sequence (xn) is uniformly distributed in [0, 1] iff for any interval I ⊂ [0, 1] : lim

N→∞

#{n ≤ N : xn ∈ I} N = |I| Equivalently, for all continuous f on [0, 1]:

1 N

N

n=1 f(xn) −

→ 1

0 f(x) dx as N → ∞.

Dmitriy Bilyk Uniform distribution: discrete vs. continuous

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SLIDE 4

Uniform distribution of sequences

A sequence (xn) is uniformly distributed in [0, 1] iff for any interval I ⊂ [0, 1] : lim

N→∞

#{n ≤ N : xn ∈ I} N = |I| Equivalently, for all continuous f on [0, 1]:

1 N

N

n=1 f(xn) −

→ 1

0 f(x) dx as N → ∞.

Weyl Criterion (1916): (xn) is uniformly distributed in [0, 1] iff for all k ∈ Z, k = 0: lim

N→∞

1 N

N

  • n=1

e2πikxn = 0

Dmitriy Bilyk Uniform distribution: discrete vs. continuous

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SLIDE 5

Uniform distribution of sequences

A sequence (xn) is uniformly distributed in [0, 1] iff for any interval I ⊂ [0, 1] : lim

N→∞

#{n ≤ N : xn ∈ I} N = |I| Equivalently, for all continuous f on [0, 1]:

1 N

N

n=1 f(xn) −

→ 1

0 f(x) dx as N → ∞.

Weyl Criterion (1916): (xn) is uniformly distributed in [0, 1] iff for all k ∈ Z, k = 0: lim

N→∞

1 N

N

  • n=1

e2πikxn = 0 The sequence {nθ} is uniformly distributed in [0, 1] iff θ is irrational.

Dmitriy Bilyk Uniform distribution: discrete vs. continuous

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SLIDE 6

Uniform distribution of sequences

A sequence (xn) is uniformly distributed in [0, 1] iff for any interval I ⊂ [0, 1] : lim

N→∞

#{n ≤ N : xn ∈ I} N = |I| Equivalently, for all continuous f on [0, 1]:

1 N

N

n=1 f(xn) −

→ 1

0 f(x) dx as N → ∞.

Weyl Criterion (1916): (xn) is uniformly distributed in [0, 1] iff for all k ∈ Z, k = 0: lim

N→∞

1 N

N

  • n=1

e2πikxn = 0 The sequence {nθ} is uniformly distributed in [0, 1] iff θ is irrational. For any subsequence (nk) of integers, the sequence {nkθ} is uniformly distributed for a.e. θ.

Dmitriy Bilyk Uniform distribution: discrete vs. continuous

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Discrepancy

Discrepancy of a sequence For a sequence ω = (ωn)∞

n=1 and an interval I ⊂ [0, 1] consider

the quantity ∆N,I = ♯{ωn : ωn ∈ I; n ≤ N} − N|I|. Define DN = sup

I⊂[0,1]

  • ∆N,I
  • .

Dmitriy Bilyk Uniform distribution: discrete vs. continuous

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SLIDE 8

Discrepancy

Discrepancy of a sequence For a sequence ω = (ωn)∞

n=1 and an interval I ⊂ [0, 1] consider

the quantity ∆N,I = ♯{ωn : ωn ∈ I; n ≤ N} − N|I|. Define DN = sup

I⊂[0,1]

  • ∆N,I
  • .

A sequence (ωn)∞

n=1 is u.d. in [0, 1]

if and only if lim

N→∞

DN N = 0.

Dmitriy Bilyk Uniform distribution: discrete vs. continuous

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Erd˝

  • s-Turan inequality

Theorem (Erd˝

  • s-Turan)

For any sequence ω ⊂ [0, 1] we have DN(ω) N m +

m

  • h=1

1 h

  • N
  • n=1

e2πihωn

  • for all natural numbers m.

Dmitriy Bilyk Uniform distribution: discrete vs. continuous

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Erd˝

  • s-Turan inequality

Theorem (Erd˝

  • s-Turan)

For any sequence ω ⊂ [0, 1] we have DN(ω) N m +

m

  • h=1

1 h

  • N
  • n=1

e2πihωn

  • for all natural numbers m.

Folklore: misses optimal estimates by a logarithm.

Dmitriy Bilyk Uniform distribution: discrete vs. continuous

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SLIDE 11

Erd˝

  • s-Turan inequality

Theorem (Erd˝

  • s-Turan)

For any sequence ω ⊂ [0, 1] we have DN(ω) N m +

m

  • h=1

1 h

  • N
  • n=1

e2πihωn

  • for all natural numbers m.

Folklore: misses optimal estimates by a logarithm. E.g., for a badly approximable irrational θ, Erd˝

  • s-Turan

yields DN({nθ}) log2 N, while in fact DN({nθ}) log N.

Dmitriy Bilyk Uniform distribution: discrete vs. continuous

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Erd˝

  • s-Turan inequality

Theorem (Erd˝

  • s-Turan)

For any sequence ω ⊂ [0, 1] we have DN(ω) N m +

m

  • h=1

1 h

  • N
  • n=1

e2πihωn

  • for all natural numbers m.

Folklore: misses optimal estimates by a logarithm. E.g., for a badly approximable irrational θ, Erd˝

  • s-Turan

yields DN({nθ}) log2 N, while in fact DN({nθ}) log N. For ω = {nθ} sharper bounds can be obtained using continued fractions.

Dmitriy Bilyk Uniform distribution: discrete vs. continuous

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Irregularities of distribution

Can discrepancy stay small? Consider a sequence ω = (ωn)∞

n=1 ⊂ [0, 1].

Dmitriy Bilyk Uniform distribution: discrete vs. continuous

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Irregularities of distribution

Can discrepancy stay small? Consider a sequence ω = (ωn)∞

n=1 ⊂ [0, 1].

van der Corput (1934): Can DN(ω) be bounded as N → ∞?

Dmitriy Bilyk Uniform distribution: discrete vs. continuous

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SLIDE 15

Irregularities of distribution

Can discrepancy stay small? Consider a sequence ω = (ωn)∞

n=1 ⊂ [0, 1].

van der Corput (1934): Can DN(ω) be bounded as N → ∞? van Aardenne-Ehrenfest (1945): NO!

Dmitriy Bilyk Uniform distribution: discrete vs. continuous

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Irregularities of distribution

Can discrepancy stay small? Consider a sequence ω = (ωn)∞

n=1 ⊂ [0, 1].

van der Corput (1934): Can DN(ω) be bounded as N → ∞? van Aardenne-Ehrenfest (1945): NO! Theorem (K. Roth, 1954) The following are equivalent: (i) For every ω = (ωn)∞

n=1 ⊂ [0, 1],

DN(ω) f(N) for infinitely many values of N. (ii) For any distribution PN ⊂ [0, 1]2 of N points, sup

R− rectangle

  • #PN ∩ R − N · |R|
  • f(N)

Dmitriy Bilyk Uniform distribution: discrete vs. continuous

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Irregularities of Distribution: simplest example

X – roll of a single die

Dmitriy Bilyk Uniform distribution: discrete vs. continuous

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Irregularities of Distribution: simplest example

X – roll of a single die

  • X − EX
  • ≥ 1

2

Dmitriy Bilyk Uniform distribution: discrete vs. continuous

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Geometric Discrepancy

PN – a set of N points in [0, 1]d R –a geometric family (e.g. axis-parallel rectangles, all rectangles, polytopes, balls, convex sets etc.)

Dmitriy Bilyk Uniform distribution: discrete vs. continuous

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Geometric Discrepancy

PN – a set of N points in [0, 1]d R –a geometric family (e.g. axis-parallel rectangles, all rectangles, polytopes, balls, convex sets etc.) Discrepancy of PN with respect to R ∈ R D(PN, R) = ♯{PN ∩ R} − N · vol (R)

Dmitriy Bilyk Uniform distribution: discrete vs. continuous

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Geometric Discrepancy

PN – a set of N points in [0, 1]d R –a geometric family (e.g. axis-parallel rectangles, all rectangles, polytopes, balls, convex sets etc.) Discrepancy of PN with respect to R ∈ R D(PN, R) = ♯{PN ∩ R} − N · vol (R) Discrepancy of PN with respect to R D(PN) = sup

R∈R

|D(PN, R)|

Dmitriy Bilyk Uniform distribution: discrete vs. continuous

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Geometric Discrepancy

PN – a set of N points in [0, 1]d R –a geometric family (e.g. axis-parallel rectangles, all rectangles, polytopes, balls, convex sets etc.) Discrepancy of PN with respect to R ∈ R D(PN, R) = ♯{PN ∩ R} − N · vol (R) Discrepancy of PN with respect to R D(PN) = sup

R∈R

|D(PN, R)| D(N) = inf

PN D(PN)

Dmitriy Bilyk Uniform distribution: discrete vs. continuous

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Discrepancy function

Consider a set PN ⊂ [0, 1]d consisting of N points: Define the discrepancy function of the set PN as DN(x) = ♯{PN ∩ [0, x)} − Nx1x2 . . . xd

Dmitriy Bilyk Uniform distribution: discrete vs. continuous

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Numerical integration

Koksma-Hlawka inequality:

  • [0,1]d f(x) dx − 1

N

  • p∈PN

f(p)

  • 1

N V (f) · DN∞

Dmitriy Bilyk Uniform distribution: discrete vs. continuous

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Numerical integration

Koksma-Hlawka inequality:

  • [0,1]d f(x) dx − 1

N

  • p∈PN

f(p)

  • 1

N V (f) · DN∞ V (f) is the Hardy-Krause variation of f

Dmitriy Bilyk Uniform distribution: discrete vs. continuous

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Numerical integration

Koksma-Hlawka inequality:

  • [0,1]d f(x) dx − 1

N

  • p∈PN

f(p)

  • 1

N fx1...xd1 · DN∞ V (f) is the Hardy-Krause variation of f V (f) =

  • [0,1]d
  • ∂df

∂x1∂x2 . . . ∂xd

  • dx1 . . . dxd

e.g., if f(x1, ..., xd) = 1

x1 ...

1

xd φ(y)dy

Dmitriy Bilyk Uniform distribution: discrete vs. continuous

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Roth’s Theorem

Klaus Roth, October 29, 1925 – November 10, 2015 Theorem (ROTH, K. F. On irregularities of distribution, Mathematika 1 (1954), 73–79.) There exists Cd ≥ 0 such that for any N-point set PN ⊂ [0, 1]d DN2 ≥ Cd(log N)

d−1 2 . Dmitriy Bilyk Uniform distribution: discrete vs. continuous

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Roth’s Theorem

According to Roth himself, this was his favorite result. William Chen (private communication) Kenneth Stolarsky (private communication) Ben Green (comment on Terry Tao’s blog)

Dmitriy Bilyk Uniform distribution: discrete vs. continuous

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Roth’s Theorem: legacy

Theorem (ROTH, K. F. On irregularities of distribution, Mathematika 1 (1954), 73–79.) There exists Cd ≥ 0 such that for any N-point set PN ⊂ [0, 1]d DN2 ≥ Cd(log N)

d−1 2 .

4 papers by Roth (On irregularities of distribution. I–IV) 10 papers by W.M. Schmidt (On irregularities of

  • distribution. I–X)

2 by J. Beck (Note on irregularities of distribution. I–II) 4 by W. W. L. Chen (On irregularities of distribution. I–IV) 2 by Beck and Chen (Note on irregularities of distribution. I–II) a book by Beck and Chen, “Irregularities of distribution”.

Dmitriy Bilyk Uniform distribution: discrete vs. continuous

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References

Books Kuypers, Niederreiter “Uniform distribution of sequences” Beck, Chen “Irregularities of distribution” Drmota, Tichy “ Sequences, discrepancies and applications” Matouˇ sek “Geometric discrepancy” Dick, Pillichshammer “Digital nets and sequences” Chazelle “Discrepancy method”

Dmitriy Bilyk Uniform distribution: discrete vs. continuous

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Average case: Lp discrepancy, 1 < p < ∞

Theorem (Roth, 1954 (p = 2); Schmidt, 1977 (1 < p < 2)) The following estimate holds for all PN ⊂ [0, 1]d with #PN = N: DNp (log N)

d−1 2 Dmitriy Bilyk Uniform distribution: discrete vs. continuous

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Average case: Lp discrepancy, 1 < p < ∞

Theorem (Roth, 1954 (p = 2); Schmidt, 1977 (1 < p < 2)) The following estimate holds for all PN ⊂ [0, 1]d with #PN = N: DNp (log N)

d−1 2

Theorem (Davenport, 1956 (d = 2, p = 2); Roth, 1979 (d ≥ 3, p = 2); Frolov, 1980 (p > 2, d = 2); Chen, 1983 (p > 2, d ≥ 3); Chen, Skriganov, 2000’s) There exist sets PN ⊂ [0, 1]d with DNp (log N)

d−1 2 Dmitriy Bilyk Uniform distribution: discrete vs. continuous

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L∞: “worst-case” discrepancy

Conjecture DN∞ ≫ (log N)

d−1 2 Dmitriy Bilyk Uniform distribution: discrete vs. continuous

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L∞: “worst-case” discrepancy

Conjecture DN∞ ≫ (log N)

d−1 2

Theorem (Schmidt, 1972; Hal´ asz, 1981) In dimension d = 2 we have DN∞ log N

Dmitriy Bilyk Uniform distribution: discrete vs. continuous

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L∞: “worst-case” discrepancy

Conjecture DN∞ ≫ (log N)

d−1 2

Theorem (Schmidt, 1972; Hal´ asz, 1981) In dimension d = 2 we have DN∞ log N d = 2: Lerch, 1904; van der Corput, 1934 There exist PN ⊂ [0, 1]2 with DN∞ ≈ log N

Dmitriy Bilyk Uniform distribution: discrete vs. continuous

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SLIDE 36

Low discrepancy sets

The irrational (α = √ 2) lattice with N = 212 points

  • n/N, {nα}
  • ,

n = 0, 1, ..., N − 1. Discrepancy ≈ log N

Dmitriy Bilyk Uniform distribution: discrete vs. continuous

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SLIDE 37

Low discrepancy sets

The van der Corput set with N = 2n points (here n = 12)

  • 0.x1x2...xn, 0.xnxn−1...x2x1
  • ,

xk = 0 or 1. Discrepancy ≈ log N

Dmitriy Bilyk Uniform distribution: discrete vs. continuous

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SLIDE 38

van der Corput set

van der Corput set with N = 23 points

Dmitriy Bilyk Uniform distribution: discrete vs. continuous

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SLIDE 39

van der Corput set

van der Corput set with N = 24 points

Dmitriy Bilyk Uniform distribution: discrete vs. continuous

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van der Corput set

van der Corput set with N = 25 points

Dmitriy Bilyk Uniform distribution: discrete vs. continuous

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van der Corput set

van der Corput set with N = 26 points

Dmitriy Bilyk Uniform distribution: discrete vs. continuous

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SLIDE 42

van der Corput set

van der Corput set with N = 27 points

Dmitriy Bilyk Uniform distribution: discrete vs. continuous

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SLIDE 43

van der Corput set

van der Corput set with N = 28 points

Dmitriy Bilyk Uniform distribution: discrete vs. continuous

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SLIDE 44

van der Corput set

van der Corput set with N = 29 points

Dmitriy Bilyk Uniform distribution: discrete vs. continuous

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SLIDE 45

van der Corput set

van der Corput set with N = 210 points

Dmitriy Bilyk Uniform distribution: discrete vs. continuous

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SLIDE 46

van der Corput set

van der Corput set with N = 211 points

Dmitriy Bilyk Uniform distribution: discrete vs. continuous

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SLIDE 47

van der Corput set

van der Corput set with N = 212 points

Dmitriy Bilyk Uniform distribution: discrete vs. continuous

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SLIDE 48

van der Corput set

Dmitriy Bilyk Uniform distribution: discrete vs. continuous

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SLIDE 49

L∞ estimates

Conjecture DN∞ ≫ (log N)

d−1 2

Theorem (Schmidt, 1972; Hal´ asz, 1981) In dimension d = 2 we have DN∞ log N d = 2: Lerch, 1904; van der Corput, 1934 There exist PN ⊂ [0, 1]2 with DN∞ ≈ log N

Dmitriy Bilyk Uniform distribution: discrete vs. continuous

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SLIDE 50

L∞ estimates

Conjecture DN∞ ≫ (log N)

d−1 2

Theorem (Schmidt, 1972; Hal´ asz, 1981) In dimension d = 2 we have DN∞ log N d = 2: Lerch, 1904; van der Corput, 1934 There exist PN ⊂ [0, 1]2 with DN∞ ≈ log N d ≥ 3, Halton, Hammersley (1960): There exist PN ⊂ [0, 1]d with DN∞ (log N)d−1

Dmitriy Bilyk Uniform distribution: discrete vs. continuous

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Conjectures and results

Conjecture 1 DN∞ (log N)d−1

Dmitriy Bilyk Uniform distribution: discrete vs. continuous

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SLIDE 52

Conjectures and results

Conjecture 2 DN∞ (log N)

d 2 Dmitriy Bilyk Uniform distribution: discrete vs. continuous

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SLIDE 53

Conjectures and results

Conjecture 2 DN∞ (log N)

d 2

Theorem (DB, M.Lacey, A.Vagharshakyan, 2008) For d ≥ 3 there exists η > 0 such that the following estimate holds for all N-point distributions PN ⊂ [0, 1]d: DN∞ (log N)

d−1 2 +η . Dmitriy Bilyk Uniform distribution: discrete vs. continuous

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SLIDE 54

Connections between problems

Dmitriy Bilyk Uniform distribution: discrete vs. continuous

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SLIDE 55

Lower and upper bounds in dimension d = 2

LOWER BOUND UPPER BOUND Axis-parallel rectangles D(N, A) log N log N D2(N, A) √log N √log N Rotated rectangles N1/4 N1/4√log N Circles N1/4 N1/4√log N Convex Sets N1/3 N1/3 log4 N

Dmitriy Bilyk Uniform distribution: discrete vs. continuous

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SLIDE 56

Geometric discrepancy

No rotations: discrepancy ≈ log N All rotations: discrepancy ≈ N1/4 (J. Beck, H. Montgomerry) Partial rotations (lacunary sets, sets of small Minkowski dimension, etc) DB, X.Ma, C. Spencer, J. Pipher (2009-2011)

Dmitriy Bilyk Uniform distribution: discrete vs. continuous

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SLIDE 57

Higher dimensions: d ≥ 3

LOWER BOUND UPPER BOUND Axis-parallel boxes L∞ (log N)

d−1 2 +η

(log N)d−1 L2 (log N)

d−1 2

(log N)

d−1 2

Rotated boxes N

1 2 − 1 2d

N

1 2 − 1 2d √log N

Balls N

1 2 − 1 2d

N

1 2 − 1 2d √log N

Convex Sets N1−

2 d+1

N1−

2 d+1 logc N Dmitriy Bilyk Uniform distribution: discrete vs. continuous

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SLIDE 58

Transference: geometric to combinatorial discrepancy

S – a set with N elements, H – a collection of subset of S, χ : S → {−1, 1} – 2-coloring (red-blue) Combinatorial discrepancy: disc(H) = min

χ max A∈H

  • x∈A

χ(x)

  • Dmitriy Bilyk

Uniform distribution: discrete vs. continuous

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SLIDE 59

Transference: geometric to combinatorial discrepancy

S – a set with N elements, H – a collection of subset of S, χ : S → {−1, 1} – 2-coloring (red-blue) Combinatorial discrepancy: disc(H) = min

χ max A∈H

  • x∈A

χ(x)

  • Combinatorial discrepancy generated by geometric systems:

Let A be a family of measurable sets and SN a set of N points. disc(SN, A) = disc

  • {SN ∩ A : A ∈ A}
  • disc(N, A) =

sup

SN⊂[0,1]d;#SN=N

disc(SN, A)

Dmitriy Bilyk Uniform distribution: discrete vs. continuous

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SLIDE 60

Transference: geometric to combinatorial discrepancy

S – a set with N elements, H – a collection of subset of S, χ : S → {−1, 1} – 2-coloring (red-blue) Combinatorial discrepancy: disc(H) = min

χ max A∈H

  • x∈A

χ(x)

  • Combinatorial discrepancy generated by geometric systems:

Let A be a family of measurable sets and SN a set of N points. disc(SN, A) = disc

  • {SN ∩ A : A ∈ A}
  • disc(N, A) =

sup

SN⊂[0,1]d;#SN=N

disc(SN, A) Lemma (S´

  • s; Beck; Lov´

asz, Spencer, Vesztergombi; ... ) Combinatorial discrepancy “is larger than” the geometric discrepancy D(N, A) ≪ disc(N, A).

Dmitriy Bilyk Uniform distribution: discrete vs. continuous

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SLIDE 61

Example: Tusn´ ady’s problem

Let Rd = {axis-parallel rectangles}. Tusn´ ady’s problem: What is the asymptotics of T(N) = disc(N, Rd) as N → ∞? d = 2: Matouˇ sek; Beck log N T(N) log5/2 N d ≥ 3: Nikolov, Matouˇ sek, 2014; Beck (log N)d−1 T(N) (log N)d+ 1

2 Dmitriy Bilyk Uniform distribution: discrete vs. continuous

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SLIDE 62

Spherical cap discrepancy

For x ∈ Sd ⊂ Rd+1, t ∈ [−1, 1] define spherical caps: C(x, t) = {y ∈ Sd : x, y ≥ t}. For a finite set Z = {z1, z2, ..., zN} ⊂ Sd define Dcap(Z) = sup

x∈Sd,t∈[−1,1]

  • #
  • Z ∩ C(x, t)
  • N

− σ

  • C(x, t)
  • .

Theorem (Beck) There exists an N-point set Z ⊂ Sd with Dcap(Z) N− 1

2 − 1 2d

log N. Theorem (Beck) For any N-point set Z ⊂ Sd Dcap(Z) N− 1

2 − 1 2d . Dmitriy Bilyk Uniform distribution: discrete vs. continuous

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SLIDE 63

Spherical caps: L2

Define the the spherical cap L2 discrepancy D(2)

cap =

 

  • Sd−1

1

−1

  • #
  • Z ∩ C(x, t)
  • N

− σ

  • C(x, t)
  • 2

dt dσ(x)  

1 2

.

Dmitriy Bilyk Uniform distribution: discrete vs. continuous

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SLIDE 64

Spherical caps: L2

Define the the spherical cap L2 discrepancy D(2)

cap =

 

  • Sd−1

1

−1

  • #
  • Z ∩ C(x, t)
  • N

− σ

  • C(x, t)
  • 2

dt dσ(x)  

1 2

. Theorem (Stolarsky invariance principle) For any finite set Z = {z1, ..., zN} ⊂ Sd−1 1 N2

N

  • i,j=1

zi−zj + cd

  • D(2)

cap

2 = const =

  • Sd−1
  • Sd−1 x − y dσ(x)dσ(y).

Dmitriy Bilyk Uniform distribution: discrete vs. continuous

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SLIDE 65

Tessellations of the sphere

x y Let x, y ∈ Sd and choose a random hyperplane z⊥, where z ∈ Sd.

Dmitriy Bilyk Uniform distribution: discrete vs. continuous

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SLIDE 66

Tessellations of the sphere

x y Let x, y ∈ Sd and choose a random hyperplane z⊥, where z ∈ Sd. Then P(z⊥ separates x and y) = P(signz, x = signz, y) = d(x, y), where d is the normalized geodesic distance on the sphere, i.e. d(x, y) = cos−1x,y

π

.

Dmitriy Bilyk Uniform distribution: discrete vs. continuous

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SLIDE 67

Hamming distance

Consider a finite set of vectors Z = {z1, z2, ..., zN} on the sphere

  • Sd. Define the Hamming distance as

dH(x, y) := #

  • zk ∈ Z : sgn(x · zk) = sgn(y · zk)
  • N

, i.e. the proportion of hyperplanes z⊥

k that separate x and y.

Dmitriy Bilyk Uniform distribution: discrete vs. continuous

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SLIDE 68

Uniform tessellations

Define ∆Z(x, y) := dH(x, y) − d(x, y).

Dmitriy Bilyk Uniform distribution: discrete vs. continuous

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SLIDE 69

Uniform tessellations

Define ∆Z(x, y) := dH(x, y) − d(x, y). Let K ⊂ Sd. We say that Z is a δ-uniform tessellation of K if sup

x,y∈K

  • ∆Z(x, y)
  • ≤ δ.

Dmitriy Bilyk Uniform distribution: discrete vs. continuous

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SLIDE 70

Uniform tessellations

Define ∆Z(x, y) := dH(x, y) − d(x, y). Let K ⊂ Sd. We say that Z is a δ-uniform tessellation of K if sup

x,y∈K

  • ∆Z(x, y)
  • ≤ δ.

Question: Given K ⊂ Sd and δ > 0, what is the smallest value of N so that there exist a δ-uniform tessellation of K by N hyperplanes?

Dmitriy Bilyk Uniform distribution: discrete vs. continuous

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SLIDE 71

Motivation

Picture from Baraniuk, Foucart, Needell, Plan, Wooters

Almost isometric embeddings of subsets of Sd.

Dmitriy Bilyk Uniform distribution: discrete vs. continuous

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SLIDE 72

Motivation

Picture from Baraniuk, Foucart, Needell, Plan, Wooters

Almost isometric embeddings of subsets of Sd. Tessellations with cells small diameter

Every cell of a δ-uniform tessellation

  • f K by hyperplanes has diameter at

most δ. If x and y are in the same cell then

d(x, y) = |d(x, y) − dH(x, y)

  • =0

| ≤ δ.

Dmitriy Bilyk Uniform distribution: discrete vs. continuous

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SLIDE 73

Motivation

Picture from Baraniuk, Foucart, Needell, Plan, Wooters

Almost isometric embeddings of subsets of Sd. Tessellations with cells small diameter

Every cell of a δ-uniform tessellation

  • f K by hyperplanes has diameter at

most δ. If x and y are in the same cell then

d(x, y) = |d(x, y) − dH(x, y)

  • =0

| ≤ δ. “One-bit” compressed sensing

Dmitriy Bilyk Uniform distribution: discrete vs. continuous

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SLIDE 74

Tessellations and discrepancy

x y Wx,y Hx = {z : z, x > 0} Wxy = Hx△Hy = {z ∈ Sd : signz, x = signz, y}

Dmitriy Bilyk Uniform distribution: discrete vs. continuous

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SLIDE 75

Tessellations and discrepancy

x y Wx,y Hx = {z : z, x > 0} Wxy = Hx△Hy = {z ∈ Sd : signz, x = signz, y} P(signz, x = signz, y) = σ(Wxy) = d(x, y)

Dmitriy Bilyk Uniform distribution: discrete vs. continuous

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SLIDE 76

Tessellations and discrepancy

x y Wx,y Hx = {z : z, x > 0} Wxy = Hx△Hy = {z ∈ Sd : signz, x = signz, y} P(signz, x = signz, y) = σ(Wxy) = d(x, y) ∆Z(x, y) = dH(x, y) − d(x, y) = #(Z ∩ Wxy) N − σ(Wxy) ∆(Z) =

  • ∆Z(x, y)
  • ∞ = sup

x,y∈Sd

  • #(Z ∩ Wxy)

N − σ(Wxy)

  • .

Dmitriy Bilyk Uniform distribution: discrete vs. continuous

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SLIDE 77

Tessellation/“Wedge” discrepancy

Lemma (DB, Lacey) There exists an N-point set Z ⊂ Sd with ∆(Z) ≤ CdN− 1

2 − 1 2d

log N.

Dmitriy Bilyk Uniform distribution: discrete vs. continuous

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SLIDE 78

Tessellation/“Wedge” discrepancy

Lemma (DB, Lacey) There exists an N-point set Z ⊂ Sd with ∆(Z) ≤ CdN− 1

2 − 1 2d

log N. Corollary This implies that for δ > 0 there exists a δ-uniform tessellation

  • f Sd by N hyperplanes with

N ≤ C′

dδ−2+

2 d+1 ·

  • log 1

δ

  • d

d+1 . Dmitriy Bilyk Uniform distribution: discrete vs. continuous

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SLIDE 79

Stolarsky principle for wedge discrepancy

Define the L2 discrepancy for wedges

  • ∆Z(x, y)
  • 2

2 =

  • Sd
  • Sd
  • 1

N

N

  • k=1

1Wxy(zk) − σ(Wxy) 2 dσ(x) dσ(y)

Dmitriy Bilyk Uniform distribution: discrete vs. continuous

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SLIDE 80

Stolarsky principle for wedge discrepancy

Define the L2 discrepancy for wedges

  • ∆Z(x, y)
  • 2

2 =

  • Sd
  • Sd
  • 1

N

N

  • k=1

1Wxy(zk) − σ(Wxy) 2 dσ(x) dσ(y) Theorem (Stolarsky principle for the tessellation of the sphere) For any finite set Z = {z1, . . . , zN} ⊂ Sd

  • ∆Z(x, y)
  • 2

2

= 1 N2

N

  • i,j=1

1 2 − d(zi, zj) 2 −

  • Sd
  • Sd

1 2 − d(x, y) 2 dσ(x) dσ(y).

Dmitriy Bilyk Uniform distribution: discrete vs. continuous

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SLIDE 81

Frame potential

Z = {z1, . . . , zN} ⊂ Sd is a frame in Rd iff there exist c, C > 0 such that for any x ∈ Rd+1 cx2 ≤

  • k

|x, zk|2 ≤ Cx2.

Dmitriy Bilyk Uniform distribution: discrete vs. continuous

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SLIDE 82

Frame potential

Z = {z1, . . . , zN} ⊂ Sd is a frame in Rd iff there exist c, C > 0 such that for any x ∈ Rd+1 cx2 ≤

  • k

|x, zk|2 ≤ Cx2. Z = {z1, . . . , zN} ⊂ Sd is a tight frame iff there exists A > 0 such that for any x ∈ Rd+1

  • k

|x, zk|2 = Ax2.

Dmitriy Bilyk Uniform distribution: discrete vs. continuous

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SLIDE 83

Frame potential

Z = {z1, . . . , zN} ⊂ Sd is a frame in Rd iff there exist c, C > 0 such that for any x ∈ Rd+1 cx2 ≤

  • k

|x, zk|2 ≤ Cx2. Z = {z1, . . . , zN} ⊂ Sd is a tight frame iff there exists A > 0 such that for any x ∈ Rd+1

  • k

|x, zk|2 = Ax2. Theorem (Benedetto, Fickus) A set Z = {z1, . . . , zN} ⊂ Sd is a tight frame in Rd+1 if and

  • nly if Z is a local minimizer of the frame potential:

F(Z) =

N

  • i,j=1

|zi, zj|2.

Dmitriy Bilyk Uniform distribution: discrete vs. continuous

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SLIDE 84

Spherical designs and Korevaar–Meyers conjecture

Z = {z1, . . . , zN} ⊂ Sd is a spherical design of order t if it generates a cubature formula, which is exact for all polynomials of degree t on Sd, i.e. 1 N

N

  • i=1

p(zi) =

  • Sd

p(z)dσ whenever deg(p) = t.

Dmitriy Bilyk Uniform distribution: discrete vs. continuous

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SLIDE 85

Spherical designs and Korevaar–Meyers conjecture

Z = {z1, . . . , zN} ⊂ Sd is a spherical design of order t if it generates a cubature formula, which is exact for all polynomials of degree t on Sd, i.e. 1 N

N

  • i=1

p(zi) =

  • Sd

p(z)dσ whenever deg(p) = t. Conjecture (Korevaar-Meyers, 1994): There exist spherical designs of order t which consist of N = O(td) points.

Dmitriy Bilyk Uniform distribution: discrete vs. continuous

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SLIDE 86

Spherical designs and Korevaar–Meyers conjecture

Z = {z1, . . . , zN} ⊂ Sd is a spherical design of order t if it generates a cubature formula, which is exact for all polynomials of degree t on Sd, i.e. 1 N

N

  • i=1

p(zi) =

  • Sd

p(z)dσ whenever deg(p) = t. Conjecture (Korevaar-Meyers, 1994): There exist spherical designs of order t which consist of N = O(td) points. Bondarenko, Radchenko, Viazovska (2012): The conjecture is true! (non-constructive)

Dmitriy Bilyk Uniform distribution: discrete vs. continuous