Conditional expanding bounds for two-variable functions over finite valuation rings

TL;DR

This paper extends sum-product bounds over finite valuation rings using spectral graph theory, showing that for certain functions and sets, the image size must grow significantly.

Contribution

It generalizes recent sum-product results to finite valuation rings and introduces bounds for two-variable functions over these rings.

Findings

Establishes lower bounds on the size of function images over finite valuation rings.

Generalizes sum-product estimates from finite fields to valuation rings.

Uses spectral graph theory methods to derive expanding bounds.

Abstract

In this paper, we use methods from spectral graph theory to obtain some results on the sum-product problem over finite valuation rings $\mathcal{R}$ of order $q^r$ which generalize recent results given by Hegyv\'ari and Hennecart (2013). More precisely, we prove that, for related pairs of two-variable functions $f(x,y)$ and $g(x,y)$, if $A$ and $B$ are two sets in $\mathcal{R}^*$ with $|A|=|B|=q^\alpha$, then \[\max\left\lbrace |f(A, B)|, |g(A, B)| \right\rbrace\gtrsim |A|^{1+\Delta(\alpha)},\] for some $\Delta(\alpha)>0$.