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arXiv · 2203.01293

Paley Graphs and S\'ark\"ozy's Theorem In Function Fields

Eric Naslund
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Abstract

S\'ark\"ozy's theorem states that dense sets of integers must contain two elements whose difference is a $k^{th}$ power. Following the polynomial method breakthrough of Croot, Lev, and Pach, Green proved a strong quantitative version of this result for $\mathbb{F}_{q}[T]$. In this paper we provide a lower bound for S\'{a}rk\"{o}zy's theorem in function fields by adapting Ruzsa's construction for the analogous problem in $\mathbb{Z}$. We construct a set $A$ of polynomials of degree $<n$ such that $A$ does not contain a $k^{th}$ power difference with $|A|=q^{n-n/2k}$. Additionally, we prove a handful of results concerning the independence number of generalized Paley Graphs, including a generalization of a claim of Ruzsa, which helps with understanding the limit of the method.

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Eric Naslund. 2022-03-02. Paley Graphs and S\'ark\"ozy's Theorem In Function Fields. https://doi.org/10.1093/qmath/haac035

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