AN OVERLAP CRITERION FOR THE TILING PROBLEM OF THE LITTLEWOOD CONJECTURE
A thesis presented to the faculty of San Francisco State University In partial fulfilment of The Requirements for The Degree
Master of Arts In Mathematics
by
Lucy Hanh Odom
San Francisco, California
August 2015 Copyright by Lucy Hanh Odom 2015 CERTIFICATION OF APPROVAL
I certify that I have read AN OVERLAP CRITERION FOR THE TILING PROBLEM OF THE LITTLEWOOD CONJECTURE by Lucy Hanh Odom and that in my opinion this work meets the criteria for approving a thesis submitted in partial fulfillment of the requirements for the degree: Master of Arts in Mathematics at San Francisco State University.
Yitwah Cheung, PhD Associate Professor of Mathematics
Federico Ardila, PhD Associate Professor of Mathematics
Serkan Hosten, PhD Professor of Mathematics AN OVERLAP CRITERION FOR THE TILING PROBLEM OF THE LITTLEWOOD CONJECTURE
Lucy Hanh Odom San Francisco State University 2015
The Littlewood Conjecture is an open problem in number theory, particularly in Diophantine approximation. In this paper, we look at the continued fractions of a pair of real numbers (α, β) that is analogous to the tiling of the plane. We offer an extension to the work already done in Samantha Lui’s thesis [3], where we will do analysis in the case where α, β Q. We will address some issues that arise in this ∈ case, and extend the definition of a tiling in [3] to work in the rational case.
I certify that the Abstract is a correct representation of the content of this thesis.
Chair, Thesis Committee Date ACKNOWLEDGMENTS
First and foremost, I would like to thank my thesis advisor, Dr. Yitwah Cheung for all his help in the process of the production of this work. As an ARCS scholarship recipient, I would also like to thank the ARCS Foundation, Northern California Chapter. Lastly, I would like to thank my friends and family for being a strong support system for me.
v TABLE OF CONTENTS
1 Introduction ...... 1
2 Encoding of A-orbits ...... 6 2.1 Λ-boxes ...... 7 2.2 Pivots and Tiles ...... 12 2.3 Connection to Littlewood Conjecture ...... 13
3 Overlap Criterion ...... 16 3.1 Sector of Tiles ...... 17
4 Subsets Closed Under ...... 21 ¬ 4.1 Lattices with Rectangular Cross Sections ...... 23
5 Results and Discussion ...... 27 5.1 Conclusion ...... 28
Appendix A: Piecewise Linear Functions ...... 32
Bibliography ...... 42
vi LIST OF FIGURES
Figure Page
1.1 Tiling for p1 = 121, p2 = 277, q = 342...... 4
1.2 Tiling for p1 = 3, p2 = 2, q = 9...... 5
3 2.1 An example of V{3} in R ...... 10
3 2.2 An example of V{1} in R ...... 10
3.1 An element a A lying in τ (u)...... 18 ∈ x 3.2 A pivot u Λ on the boundary ∂ B of the open x-face of some ∈ x a Λ-box B...... 19
4.1 A property of Σ in R2...... 21 4.2 The pivot v Λ is a minimal vector in Λ P...... 25 ∈ ∩
5.1 Tiling for p1 = 3, p2 = 1, q = 6...... 29
5.2 The line lxy with regions where the planes Px and Py lie in relation to one another...... 35
5.3 The line lxz with regions where the planes Px and Pz lie in relation to one another...... 36
5.4 The line lyz with regions where the planes Py and Pz lie in relation to one another...... 37 5.5 An illustration of the x, y, and z sectors...... 37 5.6 The set ∆(u) in the ts-plane...... 39
5.7 The contours/level sets of Γu are right angled triangles...... 40
vii 1
Chapter 1
Introduction
The Littlewood Conjecture [5] is an open problem in Diophantine approximation and was first proposed by John Edensor Littlewood in the 1930s. Take x R, and ∈ let x denote the distance from x to the nearest integer. Then for any α R there k k ∈ exists integers p, q with 1 q Q such that ≤ ≤
p 1 α , − q ≤ qQ i.e. qα 1 . The behavior of qα , as q varies through integers, embodies k k ≤ Q k k approximation of α by rational numbers. The Littlewood Conjecture handles the simultaneous approximation of two real numbers α, β. It asserts that for any two real numbers α and β, lim inf n nα nβ = 0, (1.1) n→∞ k kk k 2
where is the distance to the nearest integer. It also states that α and β may k · k be approximated, pretty well, by rational numbers with the same denominator.
It is clear that the conjecture holds under the assumptions that α, β Q. The ∈ conjecture also holds if the continued fraction of α (or β) has unbounded partial
quotients, that is, if the integers a0, a1,... with a1, a2, . . . > 0 such that
1 α = a0 + 1 a1 + 1 a2 + .. a3 + . satisfy sup a = . As a shorthand notation we write the set of partial quotients as k ∞
α = [a1; a2, a3,...]. Perhaps the most important result on the Littlewood Conjecture also happens to be the most recent. In 2006, Einsiedler et. al. [1] proved that
Theorem 1.1. The set of (α, β) for which (1.1) fails has Hausdorff dimension 0.
In [3], Lui showed that a pair of real numbers (α, β) corresponds to a Farey tiling
of the xy-plane, with the assumption that α, β R Q. One of the main results in ∈ \ [3] were inf n nα nβ > 0 n>0 k kk k
if and only if the diameters of each of the tiles are uniformly bounded. We will
continue this geometric approach to the Littlewood Conjecture when α, β Q. ∈ 3
Below is a Farey tiling of the plane for the pair (α, β) = (√2, √3). These tiles come from the partial quotients of √2 and √3 respectively. As √2 = [1; 2, 2, 2,...], and √3 = [1; 1, 2, 1, 2, 1,...], we see that the slopes of the upper boundary represent the partial quotients of √2, and the alternating slopes of the lower boundary rep- resent the partial quotients of √3. Because of the fact that the continued fractions of irrational numbers do not terminate, then naturally there are infinitely many corresponding tiles.
We now wish to understand the tilings for the pair (α, β) in the rational case, and develop an analog of the continued fraction when (α, β) Q2. Due to the finite ∈ number of partial quotients in the rational case, an implication of (α, β) Q2 is ∈ 4
that the number of tiles is finite, as can be seen by the next figure.
Figure 1.1: Tiling for p1 = 121, p2 = 277, q = 342.
p1 p2 In the following sections, we assume that the pair of rational numbers ( q , q )
p1 p2 satisfies gcd(p1, q) = 1 and gcd(p2, q) = 1, i.e., the pair ( q , q ) is totally reduced. Then we will show that the same definition for τ(u) in Lui’s thesis defines a tiling.
p1 p2 However, we will see that if ( q , q ) is not totally reduced, then Lui’s definition for the irrational case will break down. In particular, the tiles may overlap. Below is an example where gcd(p , q) = 1 and gcd(p , q) = 1. 1 6 2 6 5
Figure 1.2: Tiling for p1 = 3, p2 = 2, q = 9.
1 2 The above figure shows that the pair ( 3 , 9 ) contains two non-degenerate tiles, which are the red colored ones. The overlap takes place in a portion of the green with another portion of the black one. The portion of the green tile lies directly beneath the sliver of the upper “sector” of the the black tile. In the next sections, we will define the sectors of a tile. The goal of this project is to investigate the cause of these tiles overlapping, and establish an overlap criterion for the case of rational pairs (α, β). 6
Chapter 2
Encoding of A-orbits
To each lattice Λ Rn, we will start by defining a covering by “tiles” of the positive ⊂ diagonal subgroup A SL(n, R) given by ⊂ a1 0 0 n . Y A = 0 .. 0 : i, ai > 0, ai = 1 . ∀ i=1 0 0 an
These tiles come from the encoding of the A-orbit of Λ through the shortest nonzero vector in aΛ with respect to the sup norm ∞ as a varies over A. k · k A special case of this construction can be referred to in Appendix A, where n = 3 and a = et+s, a = et−s, a = e−2t, and the mapping (t, s) A is a homeomorphism. 1 2 3 7→ 7
2.1 Λ-boxes
Let Λ Rn be a lattice, by which we mean the discrete subgroup of Rn spanned by ⊂ n linearly independent vectors.
Definition 2.1. We define a box to be a subset of Rn of the form
n B(u) = x R : xi ui for i = 1, . . . , n { ∈ | | ≤ }
where u1, . . . , un are nonnegative real numbers, any number of which are permitted to vanish. When this happens, the box is said to be degenerate. The dimension of
B(u) refers to the dimension of the smallest subspace W Rn such that B W . ⊂ ⊂ The interior and boundary of a box B are taken relative to the subspace W and are denoted by intB and ∂B, respectively.
Remark. Note that this definition allows for degenerate boxes, i.e. a box for which
Qn n ui = 0. We say that a box B R is degenerate if and only if its dimension is i=1 ⊂ < n.
Definition 2.2. By a Λ-box, we mean a box B such that
intB Λ = 0 and ∂B Λ = ∅. ∩ { } ∩ 6
We partially order the set of all Λ-boxes by inclusion, with respect to which the notions of maximal and minimal Λ-boxes are then defined. 8
Lemma 2.1. For any non-degenerate box B Rn there is a unique a A such that ⊂ ∈ Q = aB is a cube.
Proof. For a A, we have ∈ a1 . a = .. an
such that a > 0 and Qn a = 1. If aB is a cube, then r = a u = = a u , i i=1 i 1 1 ··· n n where r is the common value of the length of the sides of the cube. It follows that
rn = (a u ) (a u ) = (a a )(u u ) = vol(B), 1 1 ··· n n 1 ··· n 1 ··· n
pn and so r = vol(B). Hence, the unique a A is the diagonal matrix with elements ∈
pn pn vol(B) vol(B) a1 = , , an = u1 ··· un that make Q = aB a cube.
Notation. Let a A and Λ R3. Then the set of all Λ-boxes is parametrized by ∈ ⊂ A via the map sending
a A B = a−1Q(aΛ) ∈ 7→ a
where Q( ) denotes the largest cube containing no nonzero lattice points in its inte- · 9
rior. As a shorthand notation, we shall write B = a−1Q.
Then one observation we can make is that
B is a Λ-box aB is an (aΛ)-cube. ⇔
If we work with boxes in R3, the box will have a multi-face of codimension 1. In general, a multi-face of codimension k has 2k components.
Definition 2.3. Given a non-degenerate box B, there is a unique a A such that ∈ Q = aB is a cube. The boundary ∂B is partitioned into the multi-faces
−1 ∂I B = a (∂Q VI ) , ∅ = I 1, , n , ∩ 6 ⊂ { ··· }
n #I where VI = x R : x ∞ = xi if and only if i I . We refer to any of the 2 { ∈ k k | | ∈ } connected components of ∂I B as a face of B of “type” I.
3 Remark. For a box in R , VI in the above definition can be geometrically thought of as a cone shaped like a pyramid with two components extending in either the x, y, or z direction, which can sometimes be labeled as 1, 2, or 3 respectively. 10
3 Figure 2.1: An example of V{3} in R .
3 Figure 2.2: An example of V{1} in R .
The next definition extends the previous one to include degenerate boxes.
Definition 2.4. Given a box B, there is a unique a A such that Q = aB is a ∈ cube of the same dimension as B and the linear map defined by a is the identity on the orthogonal complement of B. Then ∂B is partitioned in multi-faces ∂I B defined by the same formula as above, but with I restricted to nonempty subsets of the set 11
of supporting indices.
Note that the codimension of a face type I is the cardinality of I.
Notation. We shall write ∂iB as a shorthand for ∂{i}B, and we allow i to range over the set 1, . . . , n . We use to denote the set of supporting indices so that { } • ∂•B refers to the set of corners of B.
Lemma 2.2. Let B be a Λ-box. Then B is minimal if and only if the only nonzero lattice points in B lie in the corners of B, i.e. ∂B Λ ∂•B. ∩ ⊂
Proof. If ∂B contains a lattice point not in the corners, then B cannot be minimal. Conversely, if all nonzero lattice points in B lie in the corners, then any smaller box is not a Λ-box.
Definition 2.5. We say a Λ-box is relatively maximal if it is maximal with respect to the set of all Λ-boxes of dimB.
Note that a non-degenerate Λ-box is maximal if and only if it is relatively max- imal, whereas a degenerate Λ-box might be relatively maximal, but never maximal.
Lemma 2.3. Let B be a Λ-box. Then B is relatively maximal if and only if it
contains a lattice point in each codimension one face, i.e. ∂iB Λ = ∅ for each i. ∩ 6
th Proof. If ∂iB Λ = ∅, then B can be extended in the i direction to a larger Λ-box ∩ of the same dimension, and is hence not maximal. Conversely, if every codimension one face contains a lattice point, then B is relatively maximal because any box B0 12
containing B and having the same dimension will have one of these lattice points in its interior and is thus not a Λ-box.
2.2 Pivots and Tiles
Definition 2.6. We say that u Λ is a pivot of Λ if B(u) is a minimal Λ-box. The ∈ pivot u is degenerate if the box B(u) is degenerate; or equivalently, if any coordinate of u vanishes. We say u Λ is a trivial pivot if all but one component vanishes. ∈ Definition 2.7. The set of pivot classes of the lattice Λ is defined by
Π(Λ) = u Λ: B(u) is a minimal Λ-box / { ∈ } ∼
where u v if B(u) = B(v). ∼ Definition 2.8. The tile associated with a pivot u Π(Λ) is denoted by ∈
τ(u) = a A : au ∞ aw ∞ for all 0 = w Λ . { ∈ k k ≤ k k 6 ∈ }
Note that this is well-defined since u v implies au ∞ = av ∞ for all a A, so ∼ k k k k ∈ that τ(u) = τ(v).
The next theorem can be attributed to [3] in a special case.
[ Theorem 2.4. For any lattice Λ R3, we have A = τ(u). ∈ u∈Π(Λ) 13
Proof. Given a A there is a u Λ such that ∈ ∈
au ∞ = min av ∞ : 0 = v Λ , k k {k k 6 ∈ } because aΛ 0 is a closed subset of R3 0 . If u is a pivot, then a τ(u) and we \{ } \{ } ∈ n are done. If not, let Q = [ r, r] where r = au ∞. Since aΛ contains no nonzero − k k lattice points in the interior of Q, then B(au) is an (aΛ)-box, but it is not minimal, since au is not a pivot of aΛ. Let B be a minimal (aΛ)-box contained in B(au). By Lemma 2.2, B = B(au ) for some pivot u of Λ. Since B(au ) B(au) [ r, r]3, 0 0 0 ⊂ ⊂ − we have au ∞ = r. Therefore, a τ(u ) and we are done. k 0k ∈ 0
2.3 Connection to Littlewood Conjecture
Let Λα,β denote the lattice
1 0 α − 3 Λα,β := 0 1 β Z . − 0 0 1
The height of a pivot refers to the absolute value of its z-coordinate, which only depends on its pivot class. 14
Definition 2.9. Given α, β R, let ∈
π(α, β) := n Z>0 :Λα,β has a pivot of height n . { ∈ }
Remark. There are exactly two pivot classes of height zero. These are (1, 0, 0) and (0, 1, 0). The tiles corresponding to these pivots are unbounded. If α, β are both irrational, then these are the only unbounded tiles.
Remark. For Λα,β any pivot has integer height. Moreover, a pivot of nonzero height is uniquely determined by its height up to equivalence.
Theorem 2.5 ([3]). For any pair of real numbers α and β,
inf n nα nβ = inf n nα nβ n>0 k kk k n∈π(α,β) k kk k
where π(α, β) is the set of positive pivot heights in Λα,β.
Remark. Theorem 2.5 reduces the Littlewood Conjecture while considering n ∈ π(α, β).
Notation. Although we are abusing notation a bit here, we shall write τ(n) for τ(u), where n > 0 uniquely determines the pivot. 15
The Euclidean metric on R2 is transferred onto A via the isomorphism
et+s 0 0 (t, s) 0 et−s 0 . → 0 0 e−2t
which allows us to measure the size of tiles. The following gives a geometric inter- pretation of a counterexample to Littlewood’s Conjecture. The proof of this result can be found in [3].
Theorem 2.6 ([3]). For each n π(α, β), the following inequalities hold: ∈
1 ( log n nα nβ log 6) diam τ(u ) log n nα nβ . 9 − k kk k − ≤ n ≤ − k kk k
As a consequence,
inf n nα nβ > 0 if and only if sup diam τ(un) < . n>0 k kk k n∈π(α,β) ∞
Since the sup norm is not strictly convex, it is possible for distinct tiles to overlap. When both α and β are irrational, this does not happen.
Theorem 2.7 ([3]). Assume α and β are irrational. Let u and v be pivots of Λα,β such that B(u) = B(v). Then τ(u) τ(v) has empty interior. 6 ∩
We wish to extend the argument in [3] to include cases where (α, β) Q2. ∈ 16
Chapter 3
Overlap Criterion
Due to the fact that the sup norm is not strictly convex, it is possible for tiles in
τ(u) ∈ to overlap. In [3] it was shown that this does not happen for the lattice { }u Π(Λ)
Λ = Λα,β if both α and β are irrational. The issue of overlap is more delicate in the
p p case (α, β) = ( 1 , 2 ) Q2 and depends in some nontrivial way on the arithmetic q q ∈ properties of the pair.
Definition 3.1. We say that τ(u) and τ(v) overlap if int(τ(u) τ(v)) = ∅. ∩ 6
Remark. In other words, if τ(u) and τ(v) overlap, then there exists an a τ(u) τ(v) ∈ ∩ that has a neighborhood also contained in τ(u) τ(v). Note that ∩
int(τ(u) τ(v)) = int(τ(u)) int(τ(v)). ∩ ∩
3 Definition 3.2. We say (p1, p2, q) Z is a “primitive triple representing a pair ∈ 2 (α, β) Q in lowest terms” if gcd(p1, p2, q) = 1 and q > 0. ∈ 17
Remark. Here we consider the pair (α, β) = ( p1 , p2 ) to be represented by (p , p , q) q q 1 2 ∈ Z3.
Definition 3.3. We say θ Qd has uniform height if each rational coordinate ∈ has the same height. In the lowest terms of representation, θ = p , where (p, q) q ∈ d Z Z>0 and gcd(p1, . . . , pd, q) = 1. Moreover, θ has uniform height if and only if × gcd(pi, q) = 1 for each i = 1, . . . , d.
One of the main goals of this section is to establish the following theorem.
d Theorem 3.1. For any θ R , let Λθ denote the sheared integer lattice ∈ 1 θ1 − . . .. . d+1 Λθ := Z . (3.1) 1 θd − 1
Then no two tiles in τ(u): u Π(Λ ) overlap provided each coordinate of θ is { ∈ θ } irrational, i.e. θ (R Q)d, or if θ Qd and has uniform height. ∈ \ ∈ The starting point for the proof of this theorem is Lemma 3.2, but we first need to introduce the sectors of tiles τ(u).
3.1 Sector of Tiles
Consider u Π(Λ), where Λ is fixed. ∈ 18
Definition 3.4. For each u Π(Λ), the open sectors of the tile τ(u) are given by ∈
τi(u) = a intτ(u): au V{i} , ∅ = i 1, . . . , n . { ∈ ∈ } 6 ∈ { }
n where VI = x R : x ∞ = xi if and only if i I . { ∈ k k | | ∈ } We next make an important observation which bridges the connection between an element a A belonging to a sector of a tile of τ(u) with a pivot of a lattice u ∈ lying on an open face of a Λ-box B.
a τ (u) if and only if u ∂ B . ∈ i ∈ i a In the three-dimensional setting, the above observation says that an element a A belonging to the x-sector of τ(u) corresponds to a pivot u on the x-face of ∈ Λ-box. Similar for y, z cases.
Figure 3.1: An element a A lying in τ (u). ∈ x
x-sector
a A ∈
z-sector
y-sector 19
Figure 3.2: A pivot u Λ on the boundary ∂ B of the open x-face of some Λ-box ∈ x a B.
z
x-face
u Λ ∈
x
y
Lemma 3.2. Let u and v be inequivalent pivots of Λ. If τi(u) τj(v) = ∅, then ∩ 6 i = j; moreover, u and v belong to the multi-face ∂iB of some Λ-box B.
Proof. Choose any a τ (u) τ (v) and let B = a−1Q where Q is the largest cube ∈ i ∩ j in aΛ that contains no nonzero lattice points. Since a τ (u) we have au V{ } so ∈ i ∈ i that u ∂ B. Similarly, a τ (v) implies v ∂ B. Note that B is a Λ-box so that, ∈ i ∈ j ∈ j to finish the proof, it only remains to show i = j. Suppose not. Let a = a exp X (i, j) where X (i, j) is the matrix whose (i, i)- δ · δ δ entry and (j, j)-entry are δ and δ, respectively, and such that all other entries − vanish. Then for any δ > 0 we have a u ∞ < a v ∞ so that a τ(v). Since k δ k k δ k δ 6∈ a a as δ 0, this contradicts the fact that a intτ(v). δ → → ∈ Remark. This theorem implies that overlap between two tiles τ(u) and τ(v) must 20
occur in sectors of the same “type”. We do not know whether a tile can overlap with other tiles in multiple sectors.
If we study the analysis on how the tiles may overlap, we will find that there are endless possibilities in which they can overlap, some of which can lead to very complex situations. Fortunately, by Lemma 3.2, an important consequence is that tiles can only overlap in the same sectors.
Remark. Referring to Figure 1.2 in the Introduction section, the example (3, 2, 9) has two non-degenerate tiles and they overlap in their common x-sector. A portion of the x-sector of the green one lies below the x-sector of the black one.
th n Consider the i coordinate projection map πi :(Z/qZ) Z/qZ. It is not −→
hard to see that any Λθ-box contains at most one pivot class in ∂d+1B due to the
fact that a coset of ker πd+1 can only meet Λ in a shifted rectangular lattice. In the following section, we shall abstract the relevant properties of these sets, which will enable us to show that the tiles of any lattice with “rectangular cross sections” do not overlap. 21
Chapter 4
Subsets Closed Under ¬
Definition 4.1. Given u, v Rn, and i 1, . . . , n we denote by ∈ ∈ { }
u v ¬i the vector obtained from u by replacing its ith component with that of v.
Definition 4.2. Let Σ be a subset of Rn. We say that Σ is closed under , or ¬ simply -closed, if u, v Σ implies that u v Σ for each i = 1, . . . , n. ¬ ∈ ¬i ∈
Figure 4.1: A property of Σ in R2.
u = (u1, u2) (v1, u2)
(u1, v2) v = (v1, v2) 22
Example 4.1. In R2, this definition says that given any two points in Σ, the in- tersection of the horizontal line through one point, say (u1, u2) and the vertical line through the other point (v1, v2) is again in Σ (and vice versa).
Definition 4.3. Let Γ be a discrete subgroup of Rn. We say Γ is rectangular if it is generated by its elements on the coordinate axes.
Lemma 4.1. Cosets of rectangular discrete subgroups if Rn are -closed. ¬
Proof. Note that -closed sets are preserved under translations as well as the action ¬ of A. Also, a product of a -closed set in Rk and a -closed set in Rl is a -closed ¬ ¬ ¬ set in Rk+l. Finally, Zk 0 as a subset of Rl where k < l is -closed. × { } ¬
Remark. The intersection of an arbitrary number of rectangular discrete subgroups is again rectangular.
Definition 4.4. Let Σ be a subset of Rn. By a minimal vector of Σ we mean v Σ ∈ such that u Σ B(v) implies u v, i.e. B(u) = B(v). ∈ ∩ ∼
For the following result, we prove that if Σ is a -closed set, then there exists a ¬ unique minimal vector up to equivalence.
Lemma 4.2. Any -closed set Σ has at most one minimal vector up to . ¬ ∼
Proof. For any u, v Σ, there exists w Σ such that B(w) = B(u) B(v). If u and ∈ ∈ ∩ v are minimal vectors, then B(u) = B(v) = B(w) so that u v. ∼ 23
We will next attempt to make the connection between subsets closed under ¬ and Λ-boxes in order to prove a result about the case when the pair of rational
p1 p2 numbers (α, β) = ( q , q ) is totally reduced.
4.1 Lattices with Rectangular Cross Sections
Recall that a pivot of Λ is said to be trivial if it lies on a coordinate axis and
degenerate if it belongs to some coordinate hyperplane ker πi.
Definition 4.5. Let Λ be a lattice in Rn. The ith cross section of Λ refers to the discrete subgroup Γ := Λ ker π , where π is the ith coordinate projection. We say i ∩ i i
Γi is rectangular if each vector in Γi can be written as an integer combination of the
trivial pivots of Λ. We say Λ has rectangular cross sections if Γi is rectangular for each i = 1, . . . , n.
Theorem 4.3. Suppose Λ is a lattice in Rn with rectangular cross sections. Then
no two tiles in τ(u) ∈ overlap. { }u Π(Λ)
Proof. If int(τ(u) τ(v)) = ∅, then there exists i, j such that τi(u) τj(v) = ∅, ∩ 6 ∩ 6 hence by Lemma 3.2, it suffices to show that every Λ-box B contains at most one pivot class in each of its multi-faces. Let Σ = Λ H , where H is the coset of ker π i ∩ i i i
containing one of the components of ∂iB. Each pivot in ∂iB determines a minimal
vector in Σi, and distinct pivot classes determine inequivalent minimal vectors. By 24
Lemma 4.1, Σ is closed under because it is a coset of Γ , which is rectangular. i ¬ i
Lemma 4.2 now implies that ∂iB can contain at most one pivot class.
Remark. Lemma 4.2 was key to prove the above theorem, as it uses the fact that -closed sets have at most one minimal vector up to equivalence, which implies the ¬ multi-faces of a Λ-box can never contain two pivots. From this we can deduce that two tiles τ(u) and τ(v) do not overlap in the same ith sector if Λ has rectangular cross sections.
Theorem 3.1 is a corollary of Theorem 4.3, by the next lemma.
d+1 Lemma 4.4. The lattice Λθ R has rectangular cross sections if and only if ⊂ each coordinate of θ Rd is irrational or θ Qd has uniform height. ∈ ∈ d Proof. Note that Γd+1 = Z 0 is rectangular, regardless of θ. Note also that × { } d Γi Zi := (Z 0 ) ker πi and equality holds if and only if θi is irrational, in ⊃ × { } ∩
which case Γi is rectangular. Thus, Λθ has rectangular cross sections if each θi is irrational.
d If θ Q has uniform height q, then v = (0,..., 0, q) is a trivial pivot of Λθ and ∈
Γi = Zi + Zv is rectangular for each i = 1, . . . , d. Hence Λθ has rectangular cross
sections if θ Qd has uniform height. ∈ d It remains to consider θ R that have both a rational coordinate θi and an ∈
irrational coordinate θj. The existence of the latter implies that any trivial pivot of Λ has vanishing last component. Hence, no vector in Γ Z can be expressed as a θ i \ i
sum of trivial pivots, so that the cross section Γi is not rectangular. 25
p p Example 4.2. Given a three-dimensional pivot v Λ (of Λ 1 , 2 ), recall that B(v) ∈ q q is a minimal Λ-box. Let P be a plane containing the x-face of the Λ-box B(v). As v Λ was a minimal vector, the intersection of B(v) with P gives rise to a ∈ rectangular subset. Hence Λ meets the plane P in a rectangular cross section. We see that u P B(v) implies B(u) = B(v). By Theorem 4.3, the tiles of τ(u) and ∈ ∩ τ(v) do not overlap.
Figure 4.2: The pivot v Λ is a minimal vector in Λ P. ∈ ∩ z P B(v)
P B(v) ∩
x v Λ ∈
y
Example 4.3. Observe the case of our interest, the three-dimensional lattice
1 0 p1 − q p p p2 3 Λ 1 , 2 = 0 1 Z . q q q − 0 0 1 26
We will use the results thus far to develop the overlap criterion for such a lattice, which will be discussed in the following section. 27
Chapter 5
Results and Discussion
p1 p2 Recall that the pair of rational numbers (α, β) = ( q , q ) is totally reduced if
gcd(p1, q) = 1 = gcd(p2, q).
p1 p2 We previously defined the pair (α, β) = ( q , q ) having uniform height if the above condition is satisfied in the multi-dimensional case.
Lemma 5.1. If gcd(p1, q) = 1, then the lattice Λ p1 p2 meets the x = 0 plane in q , q a rectangular cross section. Similarly, if gcd(p2, q) = 1, then the lattice meets the y = 0 plane in a rectangular cross section.
p p p1 p2 3 Proof. A vector in Λ 1 , 2 is of the form (a c, b c, c) for some (a, b, c) Z . q q − q · − q · ∈ p1 a If a = c, then gcd(p1, q) = 1 implies c = kq, where k = . The vector we obtain q · p1 is then (0, b p2k, kq), which is the Z-span of (0, 1, 0) and (0, 0, q), i.e. the integer −
p p combination of the trivial pivots of Λ 1 , 2 x = 0 . Conversely, any vector in q q ∩ { } 28
Z(0, 1, 0) + Z(0, 0, q) is realized: given m(0, 1, 0) + n(0, 0, q), let c = nq, b = p2n + m,
and a = p1n. Similarly for gcd(p2, q) = 1.
Remark. Lemma 5.1 asserts that the affine subspace containing the x-face of the cube Q = aB intersects Λ p1 p2 in a rectangular lattice. There is a unique pivot of 2 , q that face by Lemma 4.2. This means there is no overlap in the x-face of the cube
p p Q = aB. This yields similar results for Λ 1 , 2 y = 0 . q q ∩ { }
5.1 Conclusion
To conclude our work, we were able to achieve the goal of developing an overlap criterion for the tiling problem of the Littlewood Conjecture. In the process, we were able to reformulate the original definition of tiling in [3] to extend into higher dimensions. The final product of this work gives us the following theorem.
d Theorem 5.2 (Overlap Criterion). For any θ R , let Λθ denote the sheared integer ∈ lattice 1 θ1 − . . .. . d+1 Λθ := Z . (5.1) 1 θd − 1
Then no two tiles in τ(u): u Π(Λ ) overlap provided each coordinate of θ is { ∈ θ } d d irrational, i.e. θ (R Q) , or if θ Q and has uniform height, i.e, gcd(pi, q) = 1, ∈ \ ∈ 29
pi and where θi = q .
Let us transition to the case of our interest, when θ R2. Strictly speaking, the ∈ first part of the theorem was proven in [3], so that there is no overlap of tiles when θ = (α, β) are irrational. The second part of the theorem states that if the rational
p1 p2 pair (α, β) = ( q , q ) has uniform height (is totally reduced), then there is no overlap of tiles. The converse is not necessarily true, that is, the nonoverlapping of a rational pair of tiles does not mean that it is totally reduced. As an example of pair that is not totally reduced, but its tiles do not overlap, consider (p1, p2, q) = (3, 1, 6).
Figure 5.1: Tiling for p1 = 3, p2 = 1, q = 6.
As in the Figure 1.2 from the Introduction section, we obtain two nondegenerate 30
tiles, which are the red colored ones. However, overlap does not occur here because the green tile is a “full” tile, in which the red tiles do not overlap in either the x- or y-sector.
For the three-dimensional lattice Λα,β, the important observation in Section 3.1 says that an element a A SL(3, R) in an open ith sector of a tile corresponds to ∈ ⊂ a pivot on an open ith face of a Λ-box. This leaves us with three things to consider. We look at the plane plane intersecting the three-dimensional Λ-box in the x-face, y-face, and the z-face. We were able to immediately rule out the fact that Λ z = 0 is a rectangular α,β ∩{ } cross section due to the nature of the sheared integer lattice
1 0 α − 3 Λα,β := 0 1 β Z . − 0 0 1
More precisely, a vector in Λ takes the form (m αn, m βn, n) for some α,β 1 − 2 − 3 (m1, m2, n) Z . If n = 0, then the vector obtained is the Z-span of (1, 0, 0) and ∈
(0, 1, 0), which means that the three-dimensional lattice Λα,β meets the plane z = 0 in a rectangular cross section. As before, rectangular cross-sections imply there is a unique pivot. Hence, the z-face of a Λ-box can never contain two pivots. A geometric consequence is that for pivots u and v, their tiles τ(u) and τ(v) cannot 31
overlap in the z-sector. The treatment of the intersections of the x = 0 plane and y = 0 plane with the lattice Λα,β was more delicate in the sense that it required the extra totally reduced condition. This case has already been taken care of by Lemma 5.1. Therefore, the totally reduced condition ensures that there is no overlapping of tiles. 32
Appendix A: Piecewise Linear Functions
We provide a precise breakdown of the formulation of the original definitions of a lattice, tiling, and sectors of a tile in the three-dimensional context. Much of this work can be found in [3]. These definitions motivated the exploration of the overlap condition for the ra- tional pair of real numbers (α, β), but once we were able to extend the definitions to higher dimensions, we found that these definitions were no longer needed. In particular, we include the computations of the different sectors of a tile. We do not use the following computations in the proofs of our results, and so we include it here as a reference.
Given α, β R, let hα,β be the shear transformation defined by ∈ 1 0 α − hα,β := 0 1 β . − 0 0 1 33
For (t, s) R2, let the scaling matrix be defined by ∈ et+s 0 0 t−s gt,s := 0 e 0 . 0 0 e−2t
In addition, let
3 Λα,β := hα,βZ .
Then a lattice is defined by
Λ := gt,sΛα,β. (5.2)
3 Fix u Z 0 , where u = (m1, m2, n) and n > 0. Then ∈ \{ }
f (t, s) := log g h u ∞, (5.3) u k t,s α,β k
where ∞ denotes the usual sup norm. Unraveling the definition of the function k · k from (5.3)
f (t, s) = log g h u u k t,s α,β k = log (et+s(m nα), et−s(m nβ), e−2tn) k 1 − 2 − k = log max(et+s m nα , et−s m nβ , e−2t n ) | 1 − | | 2 − | | | 34
we have t + s + log m1 nα | − | fu(t, s) = t s + log m2 nβ − | − | 2t + log n. − Or equivalently, t + s + log nα k k fu(t, s) = t s + log nβ − k k 2t + log n, − where we assume m1 and m2 are nearest integers to nα and nβ, respectively. That is m nα = nα , and m nβ = nβ . | 1 − | k k | 2 − | k k
In (t, s) coordinates, we define the following planes
P := t + s + log nα (5.4) x k k
P := t s + log nβ (5.5) y − k k
P := 2t + log n (5.6) z −
To see where these planes intersect in (t, s) coordinates, we exhaust all possible cases of lines by setting the planes equal to each other. 35
Px = Py : This is given by the horizontal line lxy.
t + s + log nα = t s + log nβ (5.7) k k − k k 1 nβ s = log k k (5.8) 2 nα k k
Figure 5.2: The line lxy with regions where the planes Px and Py lie in relation to one another. s
Px > Py
(x) (y) fu > fu
1 nβ s = 2 log knαk lxy k k
Px < Py
(x) (y) fu < fu t
Px = Pz : This is given by the line lxz with slope -3.
t + s + log nα = 2t + log n (5.9) k k − nα s = 3t log k k (5.10) − − n 36
Figure 5.3: The line lxz with regions where the planes Px and Pz lie in relation to one another. s
lxz
nα s = 3t log k k − − n
Px < Pz Px > Pz
(x) (z) (x) (z) fu < fu fu > fu
t
Py = Pz : This is given by the line lyz with slope +3.
t s + log nβ = 2t + log n (5.11) − k k − nβ s = 3t + log k k (5.12) n 37
Figure 5.4: The line lyz with regions where the planes Py and Pz lie in relation to one another. s
lyz
t
Py < Pz Py > Pz
(y) (z) (y) (z) fu < fu fu > fu
nβ s = 3t + log k n k
Now putting these lines together, we obtain regions where the planes Px,Py, and
Pz dominate all others. These regions also indicate where the function fu(t, s) is the largest in sectors, as seen by the following figure.
Figure 5.5: An illustration of the x, y, and z sectors.
+ lxz Px > Pz > Py “x-sector” (x) fu largest
Pz > Px > Py Px > Py > Pz “z-sector” (z) fu largest + lxy Pz > Py > Px Py > Px > Pz
“y-sector” (y) fu largest Py > Pz > Px
+ lyz 38
So far, we have been working with the two-dimensional (t, s) plane. We now consider an additional axis, the w-axis and do analysis on the three-dimensional space. Define Γ := (t, s, w): w = f (t, s) . (5.13) u { u }
From [3], it was proven that Γu is contained in the union of the three planes. To see this, we compute the intersection of Γu with the ts-plane, that is, when w = 0.
P w = 0 : t + s = log nα x ∩ { } − k k P w = 0 : t s = log nβ y ∩ { } − − k k 1 P w = 0 : t = log n z ∩ { } 2
In addition, define ∆(u) := f (t, s) 0 . (5.14) { u ≤ }
The function ∆(u) gives us a triangular-shaped cone that lies below the plane w = 0. The set of lines P w = 0 and P w = 0 have slopes -1 and +1 respectively, x ∩ { } y ∩ { } so we can observe that ∆(u) is a right triangle bounded by the union of the three planes. 39
Figure 5.6: The set ∆(u) in the ts-plane. s
+ lxz
P w = 0 x ∩ { }
1 nβ s = 2 log knαk k k + P w = 0 (t∗,( st∗∗), s∗) l z ∩ { } xy
P w = 0 y ∩ { } + lyz
t log nβ log nα − k k − k k 1 t = 2 log n
Observe that Γu has a unique global minimum which occurs at the centroid
∗ ∗ ∗ 1 knβk ∗ (t , s ). It is clear that s = 2 log knαk . The t coordinate is given by the intersection of the lines lxz and lyz.
nα nβ 3t log k k = 3t + log k k − − n n 1 nα nβ t = log k kk k −6 n2
Thus, the centroid is given as
1 nα nβ 1 nβ (t∗, s∗) = log k kk k, log k k . (5.15) −6 n2 2 nα k k
From [3], the length of the hypotenuse of the right triangle ∆(u) is the diameter 40
of ∆(u) defined by diam ∆(u) := log n nα nβ (5.16) − k kk k
Figure 5.7: The contours/level sets of Γu are right angled triangles.
+ lxz
“x-sector”
“z-sector” + (t∗, s∗) lxy
“y-sector”
+ lyz
3 Now that we have established the piecewise linear functions fu for all u Z 0 , ∈ \{ } we proceed with the definition of tiling. Define the graph of the triangle as
3 Wα,β(t, s) := log inf gt,shα,βu : u Z 0 , {k k ∈ \{ }} which is equivalent to
Wα,β(t, s) := inf fu(t, s). (5.17) u∈Z3\{0} 41
Definition 5.1. Given a pair (α, β) associated with u Z3 0 , a tile is a set ∈ \{ }
τ(u) := (t, s): W (t, s) = f (t, s) . (5.18) { α,β u }
3 Fix u Z 0 , where u = (m1, m2, n) and n > 0. We will refer to u as a pivot ∈ \{ } of the lattice Λα,β. Geometrically, a tile can be thought of as “blocks” of polygons that partition the R2 plane. Given the continued fractions of the pair of real numbers (α, β), the tiles can be indexed by the collection of pivots of Λα,β. 42
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