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Overlapping Criterion for Tiling Problem of the Littlewood Conjecture

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Overlapping Criterion for Tiling Problem of the Littlewood Conjecture 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 2 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 Vf3g in R . ........................ 10 3 2.2 An example of Vf1g in R . ........................ 10 3.1 An element a A lying in τ (u): .................... 18 2 x 3.2 A pivot u Λ on the boundary @ B of the open x-face of some 2 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 2 \ 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 2 let x denote the distance from x to the nearest integer. Then for any α R there k k 2 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!1 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 2 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 1 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 2 n [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. 2 3 Below is a Farey tiling of the plane for the pair (α; β) = (p2; p3). These tiles come from the partial quotients of p2 and p3 respectively. As p2 = [1; 2; 2; 2;:::], and p3 = [1; 1; 2; 1; 2; 1;:::], we see that the slopes of the upper boundary represent the partial quotients of p2, and the alternating slopes of the lower boundary rep- resent the partial quotients of p3. 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 2 number of partial quotients in the rational case, an implication of (α; β) Q2 is 2 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 ⊂ 80 1 9 > a1 0 0 > >B C n > <B . C Y = A = B 0 .. 0 C : i; ai > 0; ai = 1 : >B C 8 > >@ A 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 1 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 f 2 j j ≤ g 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.
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