Open Collections

Yip, Chi Hoi

University of British Columbia

This dissertation is comprised of four articles, each related to a problem in arithmetic combinatorics. In particular, we study the interaction between addition and multiplication over real numbers, integers, and finite fields, respectively. In Chapter 2, we study Erdos similarity problem in the large. We show that for each sequence of real numbers whose integer parts form a set of positive upper Banach density, one can explicitly construct such a set S ⊆ ℝ that contains no affine copy of that sequence, such that |S ∩ I| ≥ 1 − ε for every interval I ⊂ ℝ with unit length, where ε > 0 is arbitrarily small. This answers a recent question of Kolountzakis and Papageorgiou. In Chapter 3, we establish the restricted sumset analogue of a celebrated conjecture of Sárközy on additive decompositions of the set of nonzero squares over a finite field. More precisely, we show that if q > 13 is an odd prime power, then the set of nonzero squares in F𝘲 cannot be written as a restricted sumset A⨣A, extending a result of Shkredov. We also prove an analogue of Erdos-Ko-Rado theorem in a family of Cayley sum graphs. In Chapter 4, we make significant progress towards a conjecture of Sarkozy on multiplicative decompositions of shifted multiplicative subgroups. In particular, we prove that for almost all primes p, the set {x² − 1 : x ∈ 𝔽*𝚙} \ {0} cannot be decomposed as the product of two sets in Fp non-trivially. In Chapter 5 and Chapter 6, we study the quantity M𝚔(n), the largest size of a generalized Diophantine tuple with property D𝚔(n), that is, a subset of positive integers such that each pairwise product is n less than a perfect k-th power. We provide a few new upper bounds on M𝚔(n), and in particular they significantly improve several results by Bérczes-Dujella-Hajdu-Luca, Bhattacharjee-Dixit-Saikia, and Dixit-Kim-Murty. One main ingredient in our proof is to study the analogue of Diophantine tuples over finite fields. We also study bipartite analogues of Diophantine tuples over integers and finite fields, and refine a few results of Hajdu and Sárközy.

Text

2024-07-06

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/88587

Mathematics

University of British Columbia

UBCV

http://creativecommons.org/licenses/by-nc-nd/4.0/