On $3$-Term Arithmetic Progressions in Product Sets over Finite Rings via Spectra of Graphs

Abstract

Given two sets $A,B\subset {\mathbb{F}}_q$, of elements of the finite field ${\mathbb{F}}_q$, of $q$ elements, Shparlinski (2008) showed that the product set $\mathcal{A}\mathcal{B}=\{ab\mid a\in \mathcal{A},b\in \mathcal{B}\}$ contains an arithmetic progression of length $k\ge 3$ provided that \(k

3\) is the characteristic of $\mathbb{F}$, and $|\mathcal{A}||\mathcal{B}|\ge 2q^{2-1/(k-1)}$. In this paper, we recover Shparlinski’s result for the case of 3-term arithmetic progressions via spectra of product graphs over finite fields. We also illustrate our method in the setting of residue rings. Let $m$ be a large integer and $\mathbb{Z}/m\mathbb{Z}$ be the ring of residues mod $m$. For any two sets $\mathcal{A},\mathcal{B}\subset \mathbb{Z}/m\mathbb{Z}$ of cardinality $$|\mathcal{A}||\mathcal{B}|>m(\frac{r(m)m}{r(m{)}^{\frac{1}{2}}+1})$$, the product set $\mathcal{A}\mathcal{B}$ contains a $3$-term arithmetic progression, where $r(m)$ is the smallest prime divisor of $m$ and $r(m)$ is the number of divisors of $m$. The spectral proofs presented in this paper avoid the use of character and exponential sums, the usual tool to deal with problems of this kind.