SYMBOLIC CHARACTERIZATION OF COUNTEREXAMPLES TO LITTLEWOOD’S CONJECTURE A thesis presented to the faculty of San Francisco State University ^ 5 * In partial fulfilment of The Requirements for ^ Q \ g The Degree H w r ti . ^ 5 7 - Master of Arts In Mathematics by Miguel Cardoso San Francisco, California May 2018 I Copyright by Miguel Cardoso CERTIFICATION OF APPROVAL I certify that I have read SYMBOLIC CHARACTERIZATION OF COUN­ TEREXAMPLES TO LITTLEWOOD’S CONJECTURE by Miguel Cardoso and that in my opinion this work meets the criteria for ap­ proving a thesis submitted in partial fulfillment of the requirements for the degree: Master of Arts in Mathematics at San Francisco State Uni­ versity C U m . : Alexander Schuster Professor of Mathematics "Chun-kit Lai Associate Professor of Mathematics SYMBOLIC CHARACTERIZATION OF COUNTEREXAMPLES TO LITTLEWOOD’S CONJECTURE Miguel Cardoso San Francisco State University 2018 In the 1930’s, John Edensor Littlewood proposed the Littlewood conjecture. The conjecture states that given any two real numbers a and /3, in fJt>o n||na||\\n(3\\ = 0, where ||.|| is the distance to the nearest integer. This problem is currently unsolved in mathematics. This paper will start by establishing a framework for any collection of real numbers. We start by discussing a group of diagonal matrices acting on lattices that lead to a tiling interpretation of the Littlewood conjecture. The Roll­ back theorem is then developed so we can relate the lattices of the real numbers from the Littlewood conjecture to the lattices of their pivots. This is followed by defining windows and boomerangs, which use the Rollback theorem to get an ordering on these structures and their relationships to tiling. A counterexample to the Littlewood conjecture is then characterized using nearly-nested sequences of windows. This characterization is left for further refining. I certify that the Abstract is a correct representation of the content of this thesis. ACKNOWLEDGMENTS I would like to thank Dr. Cheung for his guidance, patience, encour­ agement and understanding. Dr. Schuster and Dr. Lai, thank you for being willing to be on my committee and for such a quick turn around on a last minute project. Jon, Greg, Lisa, George, and Donald, thank you for always being willing to talk about and work through problems with this project. Nina, thank you for your mentorship and friendship as I adjusted to the new school and the Master’s program. Gina, Logan, Corey, and everyone else I worked with, thank you for working together with me and helping me keep my math skills sharp. Thank you to the Math department at SFSU for giving me this opportunity. And thank you to my family and friends for giving me your support through this. TABLE OF CONTENTS 1 Introduction.................................................... 1 2 A-action on the Space of Lattices............. 5 2 .1 The A group.......................................... 5 2.2 L a ttic e s ................................................ 8 3 Roll-Back Theorem ....................................... 10 3.1 Balance T im e ....................................... 10 3 .2 Norms of Linear M a p s ....................... 13 4 W indows.......................................................... 19 5 Encoding Windows from a Counterexample 26 6 Conclusion....................................................... 34 Bibliography ....................................................... 36 vi 1 Chapter 1 Introduction In the 1930’s, Littlewood made a conjecture that is yet unproven. It refers to how close the integer multiple of two real numbers can get to being an integer itself. The conjecture is stated as follows: Conjecture 1 .1 . For all a, € K. inf n||?Ta||||n/3|| = 0 n>0 where ||.|| is the distance to the nearest integer. It is already known that if either a E Q or /3 G Q, the conjecture holds. This is true bccausc if a = % with p, q £ Q, then ||n^| = 0 whenever n is a multiple of q. And since n = q < oo and \\qfi\\ < oo, we have the infimum of the product is 0 . For example, if a = |, then ||n||| = 0 whenever n is a multiple of 3. So the product is 0 for any multiples of 3, making the infimum 0. 2 It has also been determined that if the continued fraction of either number has unbounded partial quotients, the conjecture holds. This means that if a — a0 + ^ - •• •]— and sup = oo. then the conjecture is satisfied. [1]. 1 ” 2 + ^: This means that conjecture 1.1 holds for uncountably many irrationals. In fact, it was proven that the set of exceptions to conjecture 1.1 has Hausdorff dimension 0 . [2], We are inspired by the MA thesis of Samantha Lui [3]. In her thesis, Lui defines a tiling of the plane for each (a, /3) so that having uniformly bounded tiles implies that we have a counterexample. It was already known that having the counterexample implies that these tiles arc uniformly bounded, so this statement bccamc if and only if. To follow this, the MA thesis of Lucy Odom [5] showed that these tiles do not overlap. In this paper, we hope to further explore this topic. However, we will set up framework for the more general conjecture: Conjecture 1 .2 . For all 9 = (91, #2 , • • •, 9d) € M.d d m fn J J | K | | = 0 where | |. 11 is the distance to the nearest integer. Again, conjecture 1.2 is known to hold if any 9{ is rational for the same reason that conjccturc 1.1 holds. In fact, conjccturc 1.1 is just the spccial ease of conjecture 3 1.2 where d — 2. This paper will focus on an arbitrary d dimensions, only reverting back to the case where d = 2 for example and when necessary. In section 2 , we define the A group. We then show how this group acts on lattices. From there, we discuss the A + and A_ semigroups and their relation to the Littlewood conjecture. Section 3 start by discussing balance time. This special element of the A group depends on both the particular 9 and rational point v e R d+1 that we are looking at. We determine its uniqueness in this section as well. From there section 3 continues on to develop the Roll-back theorem. This the­ orem relates the linear maps from the A group to the lattices involved in the Lit­ tlewood conjecture. More precisely, the Roll-back theorem gives us relationships between the lattices of d and v, along with the A + and A _ groups using the balance time. In section 4, we return to focusing on the case where d = 2 . We first define a window, which has rational endpoints and arise naturally from domains of approx­ imation. This, along with the Roll-back theorem, leads us to define boomerangs. We end this section by determining how to take a sequence of windows with special properties and return a counterexample to the Littlewood conjecturc. Section 5 focuses on encoding information about a counterexample to the Little­ wood conjecture when d = 2 in windows. We determine how to create the sequence of windows that converge down to the counterexample. We also determine a list of 4 properties that arise naturally from this process. This section concludes by show­ ing that our process yields the same list of special properties assumed in the final theorem of section 5. Finally, chapter 6 concludes our work. We discuss the results, as well as future work that can be done after this thesis. 5 Chapter 2 A-action on the Space of Lattices 2.1 The A group Wc start by defining the group A. Definition 2 .1 (The Group A). In a d + 1 dimensional space, define \ 1 ai 0 ... 0 0 0 . 0 0 A = < : aj = 1 and o* > 0 for * = 1 , 2 ,..., d, d + 1 > i=i 0 o .. 0 < I 0 0 .. 0 ^d+ 1 ) j Note that the condition of n»=i a« — 1 means that one term can be written as a product of the others. This means we are in a d-dimensional space. This condition and the fact that all of the a* are positive leads to another definition. 1 6 Definition 2 .2 . We put coordinates on A by defining / \ \ ( efl 0 .. 0 0 0 e*2 .. 0 0 d+1 II < o M t= 1 0 0 .. etd 0 * gtd+i V 0 0 .. 0 J J where U = log (a.,). We call (t\, #2, • • •, td+1) the homogeneous coordinates. In the case d = 2, we use (t, s)-coordinatcs given by \ ( et+s 0 0 ^ SO CO 0 o 1 < 03 :t,s G R > (M o o 1 < > Then the sum of the exponents is 0 as desired, and we can look at the 2- dimensional plane to learn about these structures. We now look at two important semigroups of our group A. Definition 2.3 (A+). A + = {a E A: min(a1,a2,..., ad) > 1 > a^+i} 7 We call this semigroup the strongly unstable semigroup. This semigroup has a very strict condition that the last coordinate in our diagonal matrix must be less than or equal to 1 , while all other entries of these matrices must be greater than or equal to 1 . When d = 2 , this is equivalent to 0 < |.s| < t. Now we look at a semigroup with a less strict condition. Definition 2.4 (A_). A - — {a 6 A : a^+i > max(a1: a2,..., a^) Wc call this semigroup the stable semigroup. This is the semigroup where the last term dominates the others. However, it is not as strict because we only condition that the last term is biggest with no other restrictions on the other terms. When d = 2. this is equivalent to 0 < |.s| < —31. These two semigroups lead us to the following theorem: T h eorem 2 .1 .