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 \(\mathcal{AB} = \{ab \mid a \in \mathcal{A}, b \in \mathcal{B}\}\) contains an arithmetic progression of length \(k \geq 3\) provided that \(k

3\) is the characteristic of \(\mathbb{F}\), and \(|\mathcal{A}||\mathcal{B}| \geq 2q^{2-1/(k-1)}\). 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 \(\mathcal{AB}\) 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.