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Visualizing Harmonic analysis: roots of exponential polynomials ICERM June 18, 2014 Sinai Robins Based on some joint work with Nick Gravin, Dmitry Shiryaev, and Mihalis Kolountzakis Wednesday, June 18, 14 Outline Part I. History of tilings

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Visualizing Harmonic analysis: roots of exponential polynomials

Sinai Robins Based on some joint work with Nick Gravin, Dmitry Shiryaev, and Mihalis Kolountzakis

ICERM June 18, 2014

Wednesday, June 18, 14

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Outline

Part I. History of tilings (1-tilings) Part II. Multi-tilings (k-tilings), recent results Part III. Harmonic analysis approaches/ideas

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Part I. History of tilings (1-tilings)

What kind of tilings?

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Part I. History of tilings (1-tilings)

What kind of tilings? We fix one object

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Part I. History of tilings (1-tilings)

What kind of tilings? We fix one object

?

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Part I. History of tilings (1-tilings)

What kind of tilings? We fix one object

? ?

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Part I. History of tilings (1-tilings)

What kind of tilings? We fix one object

Voderberg tiling

?

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The Hirschhorn tiling

(Michael Hirschhorn, 1976, UNSW)

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Part I. History of tilings (1-tilings)

  • 1. We fix one object
  • 2. Even more, we focus on translational tilings
  • 3. Finally, we invoke the assumption that
  • ur object is convex.

What kind of tilings do we study here?

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Part I. History of tilings (1-tilings)

What kind of tilings?

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So we consider translations by one convex object P (necessarily a polytope), and we tile Euclidean space by a set of discrete translation vectors Λ, so that (almost) every point gets covered exactly once.

Example.

This Fedorov solid (also known as a Rhombic Dodecahedron) tiles R3

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Indicator functions

Definition. Given any set P ⊂ Rd, we define 1P (x) := ( 1 if x ∈ P if x / ∈ P.

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So to be a bit more Bourbaki about it, we may write:

Definition. We say that P tiles Rd with the discrete multi set of vectors Λ if X

λ∈Λ

1P +λ(v) = 1, for all v / ∈ ∂P + Λ.

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Question 1. What is the structure of a polytope P that tiles all of Euclidean space by translations, with some discrete set of vectors Λ? For example, when is it a zonotope? What do its facets look like?

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Question 1. What is the structure of a polytope P that tiles all of Euclidean space by translations, with some discrete set of vectors Λ? Question 2. What is the structure of the discrete set of vectors Λ? For example, when is it a zonotope? What do its facets look like? For example, does Λ have to be a lattice? When? Why? Can Λ be a finite union of lattices?

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1-tilings in R3

Nikolai Fyodorovich Fedorov

  • Theorem. (Fedorov, 1885) There are 5 different combinatorial types of con-

vex bodies that tile R3.

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What about higher dimensions? Can we “classify” all polytopes that tile Rd by translations?

1-tilings in Rd

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Minkowski gave necessary conditions for a polytope to tile Rd.

The first results for tiling Euclidean space in general dimension were given by Hermann Minkowski.

Minkowski gives a partial answer

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Minkowski’s result

  • Theorem. (Minkowski, 1897)

If a convex polytope P tiles Rd by translations, then:

  • 1. P must be centrally symmetric
  • 2. Each facet of P must be centrally symmetric

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  • Corollary. Every polytope that tiles R1, R2, or R3

by translations is a zonotope.

Minkowski’s result

  • Theorem. (Minkowski, 1897)

If a convex polytope P tiles Rd by translations, then:

  • 1. P must be centrally symmetric
  • 2. Each facet of P must be centrally symmetric

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  • Corollary. Every polytope that tiles R1, R2, or R3

by translations is a zonotope.

Minkowski’s result

  • Theorem. (Minkowski, 1897)

If a convex polytope P tiles Rd by translations, then:

  • 1. P must be centrally symmetric
  • 2. Each facet of P must be centrally symmetric

What’s that?

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Zonotopes

Definition. A Zonotope is a polytope P with the following equivalent properties:

  • 1. All of the faces of P are centrally symmetric
  • 2. P is the Minkowski sum of a finite number of line-segments
  • 3. P is the affine image of some n-dimensional cube [0, 1]n.

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A zonotope with 9 generators

Example.

This is the projection of a 9-dimensional cube into R3

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The 24-cell, a source of counterexamples

The 24-cell is a 4-dimensional polytope, arising as the Voronoi cell of the lattice D4 ⊂ R4.

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The 24-cell, a source of counterexamples

The 24-cell is a 4-dimensional polytope, arising as the Voronoi cell of the lattice D4 ⊂ R4.

It tiles R4 but it is not a zonotope.

The lattice D4 is defined by: D4 := {x ∈ Zd | Pd

k=1 xk ≡ 0 mod 2}

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The 24-cell, a source of counterexamples

The 24-cell is a 4-dimensional polytope, arising as the Voronoi cell of the lattice D4 ⊂ R4.

It tiles R4 but it is not a zonotope.

  • Quiz. why not?

The lattice D4 is defined by: D4 := {x ∈ Zd | Pd

k=1 xk ≡ 0 mod 2}

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The 24-cell, a source of counterexamples

The 24-cell is a 4-dimensional polytope, arising as the Voronoi cell of the lattice D4 ⊂ R4.

It tiles R4 but it is not a zonotope.

  • Quiz. why not?
  • Answer. It has a face which is not centrally symmetric.

The lattice D4 is defined by: D4 := {x ∈ Zd | Pd

k=1 xk ≡ 0 mod 2}

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  • Def. A Voronoi cell (at the origin) of any lattice

L is defined to be {x ∈ Rd | d(x, 0) ≤ d(x, l), for all l ∈ L} This region is, almost by definition, a polytope (why?).

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  • Def. A Voronoi cell (at the origin) of any lattice

L is defined to be {x ∈ Rd | d(x, 0) ≤ d(x, l), for all l ∈ L} This region is, almost by definition, a polytope (why?).

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  • Def. A Voronoi cell (at the origin) of any lattice

L is defined to be {x ∈ Rd | d(x, 0) ≤ d(x, l), for all l ∈ L} This region is, almost by definition, a polytope (why?).

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The Venkov-McMullen result, a converse to Minkowski

After 50 years passed, a converse to Minkowski’s Theorem was found.

  • Theorem. (Minkowski, 1897; Venkov, 1954; McMullen, 1980)

A convex polytope P tiles Rd by translations if and only if:

  • 1. P is centrally symmetric
  • 2. Each of the facets of P is centrally symmetric.
  • 3. Each belt of P contains either 4 or 6 codimension 2 faces.

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The Venkov-McMullen result, a converse to Minkowski

After 50 years passed, a converse to Minkowski’s Theorem was found.

  • Theorem. (Minkowski, 1897; Venkov, 1954; McMullen, 1980)

A convex polytope P tiles Rd by translations if and only if:

  • 1. P is centrally symmetric
  • 2. Each of the facets of P is centrally symmetric.
  • 3. Each belt of P contains either 4 or 6 codimension 2 faces.

In R4: 52 distinct tiling polytopes In R5: a few thousand. . . .

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The Venkov-McMullen result, a converse to Minkowski

After 50 years passed, a converse to Minkowski’s Theorem was found.

  • Theorem. (Minkowski, 1897; Venkov, 1954; McMullen, 1980)

A convex polytope P tiles Rd by translations if and only if:

  • 1. P is centrally symmetric
  • 2. Each of the facets of P is centrally symmetric.
  • 3. Each belt of P contains either 4 or 6 codimension 2 faces.

In R4: 52 distinct tiling polytopes In R5: a few thousand. . . .

What’s that?

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Example.

The red belt for this zonotope consists of 8 faces (1-dimensional faces).

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Example.

The red belt for this zonotope consists of 8 faces (1-dimensional faces).

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Example.

The red belt for this zonotope consists of 8 faces (1-dimensional faces).

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Example.

The red belt for this zonotope consists of 8 faces (1-dimensional faces).

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Example.

The red belt for this zonotope consists of 8 faces (1-dimensional faces).

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Example.

The red belt for this zonotope consists of 8 faces (1-dimensional faces).

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Example.

The red belt for this zonotope consists of 8 faces (1-dimensional faces).

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Example.

The red belt for this zonotope consists of 8 faces (1-dimensional faces).

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Example.

The red belt for this zonotope consists of 8 faces (1-dimensional faces).

This polytope therefore does not tile R3 by translations, since it violates condition (3) of the Venkov-McMullen Theorem.

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  • Example. Does this one tile by translations?

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Example.

Yes!

Does this one tile by translations?

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Example.

Another construction for this Fedorov solid is obtained by truncating the octahedron. Yet another construction for it is obtained by considering it as a Permutahedron in R4

Yes!

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Part II. Multi-tilings (k-tilings)

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Tiling with multiplicities

A natural generalization of a tiling is a tiling with multiplicity k. (also called a k-tiling, or a multi-tiling)

X

λ∈Λ

1P +λ(v) = k,

Definition. We say that a polytope P tiles Rd with a discrete set of translation vectors Λ if for all v / ∈ ∂P + Λ.

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  • Example. The integer octagon

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2-dimensional results for k-tilings

1994: Bolle gave a nice combinatorial characterization of all lattice k- tilings of R2 in terms of distances between vertices of a polygon.

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2-dimensional results for k-tilings

1994: Bolle gave a nice combinatorial characterization of all lattice k- tilings of R2 in terms of distances between vertices of a polygon.

2000: Kolountzakis proved that for every k-tiling of R2 with a multiset Λ, Λ must be a finite union of lattices.

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2-dimensional results for k-tilings

1994: Bolle gave a nice combinatorial characterization of all lattice k- tilings of R2 in terms of distances between vertices of a polygon.

2000: Kolountzakis proved that for every k-tiling of R2 with a multiset Λ, Λ must be a finite union of lattices.

2013: Dmitry Shiryaev has recently shown (Ph.d thesis) that in R2 every k-tiler must in fact tile with one lattice (i.e. must be periodic).

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A structure theorem for d-dimensional polytopes that multi-tile

Theorem.(Gravin, R., Shiryaev, Combinatorica, 2012) Suppose a polytope P multiply-tiles Rd with a discrete multiset L. Then P is symmetric, and each facet of P is also symmetric.

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A partial converse

Suppose that a polytope P enjoys the following properties:

  • 1. P is symmetric
  • 2. Each facet of P is also symmetric
  • 3. P is a rational polytope (all vertices are rational points).

Then P multi-tiles with the integer lattice Zd.

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Suppose P k-tiles Rd with the set of translation vectors Λ. Then for every general position of -P, there are exactly k points of Λ in the interior of -P.

Technique: counting Λ-points inside P

(“standing at v and looking at λ” versus “standing at λ and looking at v)

Easy proof: X

λ∈Λ

1−P +v(λ) = X

λ∈Λ

1P +λ(v) = k, because λ ∈ −P + v if and only if v ∈ P + λ.

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Solid angles (volumes of spherical polytopes) play an equivalent role, too!

The previous simple observation has an interesting extension. Let ωP (x) be the proportion of P which intersects a small sphere centered at x ∈ Rd. It’s also called a solid angle at x, relative to P.

  • Theorem. (2013, Gravin, R, Shiryaev) A polytope P k-tiles Rd with

the discrete set of translations Λ if and only if

P

λ∈Λ ωP +v(λ) = k,

for all v ∈ Rd.

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  • Example. An integer octagon that 7-tiles R2.
  • Example. An integer octagon that 7-tiles R2.

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Part III. Harmonic analysis approach/ideas

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A structure theorem for the 3-dimensional set of translation vectors

Theorem.(Gravin, Kolountzakis, R., Shiryaev, 2013, to appear in Discrete and Computational Geometry) Suppose a polytope P multiply-tiles with a discrete multiset L, and suppose that P is not a two-flat zonotope. Then L is a finite union of translated lattices.

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A structure theorem for the 3-dimensional set of translation vectors

(Proof uses the idempotent theorem in Fourier analysis, due to Meyer and later developed by Paul Cohen.)

Theorem.(Gravin, Kolountzakis, R., Shiryaev, 2013, to appear in Discrete and Computational Geometry) Suppose a polytope P multiply-tiles with a discrete multiset L, and suppose that P is not a two-flat zonotope. Then L is a finite union of translated lattices.

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A two-flat zonotope with 9 generators Example.

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A two-flat zonotope with 9 generators Example.

We discovered it by playing with the formulas for the Fourier transform of polytopes.

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(This is an easy Lemma, but already shows a distinct approach)

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x cos(x) = y cos(y)

Example.

The real zero set of the Fourier transform of the square [0,1]^2 (w.r.t. uniform measure)

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Harmonic Analysis approach

Thus, we can study the vanishing of the Fourier transform of a polytope, namely ˆ 1P (m) = 0. The vanishing of Fourier transforms of convex bodies in general has been studied, in the context of the Fuglede conjecture, by Alex Iosevich, Mihalis Kolountzakis, Mate Matolci, Izabella Laba, Terry Tao, and others.

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Harmonic Analysis approach

Some other open questions:

  • 1. Give an analogue of the Venkov-McMullen converse for k-tilings.

2. Given k, describe all polytopes that k-tile. What is the smallest non- trivial k that is possible in dimension d?

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Harmonic Analysis approach

Some other open questions:

  • 1. Give an analogue of the Venkov-McMullen converse for k-tilings.

2. Given k, describe all polytopes that k-tile. What is the smallest non- trivial k that is possible in dimension d?

  • 3. Find the number of vertices of a k-tiler.

4. Most importantly for us: Using the vanishing set (as a subset of Rd)

  • f the Fourier transform ˆ

1P (m), classify all k-tiling polytopes. Focus on d = 2 first.

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Harmonic Analysis approach

Something that we keep seeing is that it’s very fruitful to simultaneously think about the Fourier analysis and the Discrete/Combinatorial geometry.

Nick Gravin, Mihail Kolountzakis, Sinai Robins, and Dmitry Shiryaev, Struc- ture results for multiple tilings in 3D, Discrete & Computational Geometry, (2013), 50, 1033-1050. Nick Gravin, Sinai Robins, and Dima Shiryaev, Translational tilings by a polytope, with multiplicity, Combinatorica 32 (6), (2012) 629-648.

References

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