Two topics in arithmetic combinatorics

This thesis - naturally delineated into two parts - attempts to address two flavors of problems in arithmetic combinatorics. These two parts are based on the papers \cite{JRT} and \cite{BFR}, respectively - all mathematical content in this thesis has already appeared in these papers (and associated preprints). In the first, together with Jing and Tran \cite{JRT} we find inspiration in the sum-product phenomenon and the classification of two-variable polynomials of bounded growth. For a bivariate $P(x,y) \in \RR[x,y]\setminus (\RR[x] \cup \RR[y])$, our first result shows that for all finite $A \subseteq \RR$, |P(A,A)|\geq α|A|^5/4 with α =α(\deg P) \in \RR^>0 unless $ P(x,y)=f(γ u(x)+δ u(y)) \text{ or } P(x,y)=f(u^m(x)uⁿ(y)) $ for some univariate $f, u \in \RR[t]\setminus \RR$, constants $\gamma, \delta \in \RR^{\neq 0}$, and $m, n\in \NN^{\geq 1}$. This resolves the symmetric nonexpanders classification problem proposed by de Zeeuw. Our second and third results in this chapter are sum-product type theorems for two polynomials, generalizing the classical result by Erd\H os and Szemer\'edi as well as a theorem by Shen. We also obtain similar results for $\CC$, and from this deduce results for fields of characteristic $0$ and fields of large prime characteristic. We use tools from semialgebraic/o-minimal geometry to prove these results; exposition is provided on these methods to make them accessible to readers primarily concerned with combinatorics. In the second, together with Balogh and F\"uredi \cite{BFR}, we combine two elementary proofs to show that the maximum size of a Sidon set of $\{ 1, 2, \ldots, n\}$ is at most $\sqrt{n}+ 0.998n^{1/4}$ for sufficiently large $n$ - the first non-constant improvement of the error term $n^{1/4}$ in this classical combinatorial number theory problem since 1969. This implies improvements in some related problems which are also discussed.

Name thesis:degree_name

Ph.D.

Level thesis:degree_level

Dissertation

Discipline thesis:degree_discipline

Mathematics

Grantor

University of Illinois at Urbana-Champaign

Year dc:date

2023

Author dc:creator

• Roy, Souktik

Contributors dc:contributor

• Balogh, Jozsef
• Kostochka, Alexandr
• Ford, Kevin
• Bradshaw, Peter

• Sum Product
• O-minimality
• Sidon Sets

Statement dc:rights

• Copyright 2023 Souktik Roy

Language dc:language

en, eng

Handle dc:identifier

https://hdl.handle.net/2142/120122

OAI record oai:identifier

https://www.ideals.illinois.edu/oai-pmh?verb=GetRecord&metadataPrefix=etdms&identifier=oai%3Awww.ideals.illinois.edu%3A2142%2F120122