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Topics in arithmetic combinatorics Yip, Chi Hoi
Abstract
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 ErdosKoRado 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 nontrivially. 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 kth power. We provide a few new upper bounds on M𝚔(n), and in particular they significantly improve several results by BérczesDujellaHajduLuca, BhattacharjeeDixitSaikia, and DixitKimMurty. 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.
Item Metadata
Title 
Topics in arithmetic combinatorics

Creator  
Supervisor  
Publisher 
University of British Columbia

Date Issued 
2024

Description 
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 ErdosKoRado 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 nontrivially.
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 kth power. We provide a few new upper bounds on M𝚔(n), and in particular they significantly improve several results by BérczesDujellaHajduLuca, BhattacharjeeDixitSaikia, and DixitKimMurty. 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.

Genre  
Type  
Language 
eng

Date Available 
20240706

Provider 
Vancouver : University of British Columbia Library

Rights 
AttributionNonCommercialNoDerivatives 4.0 International

DOI 
10.14288/1.0444095

URI  
Degree  
Program  
Affiliation  
Degree Grantor 
University of British Columbia

Graduation Date 
202411

Campus  
Scholarly Level 
Graduate

Rights URI  
Aggregated Source Repository 
DSpace

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AttributionNonCommercialNoDerivatives 4.0 International