Sums and products along sparse graphs

TL;DR

This paper explores sum-product problems on sparse graphs, especially matchings, showing how bounds in these settings relate to classical problems and providing new bounds using elliptic curve analysis.

Contribution

It establishes a connection between sum-product bounds on sparse graphs and the Erdős-Szemerédi problem, introducing new bounds and a reduction to a problem involving perfect squares.

Findings

Lower bounds on sum-product for matchings imply bounds for the original problem.

Exact bounds for sums along expanders are provided.

A reduction to a problem involving perfect squares is developed, with new results using elliptic curves.

Abstract

In their seminal paper from 1983, Erd\H{o}s and Szemer\'edi showed that any $n$ distinct integers induce either $n^{1+\epsilon}$ distinct sums of pairs or that many distinct products, and conjectured a lower bound of $n^{2-o(1)}$. They further proposed a generalization of this problem, in which the sums and products are taken along the edges of a given graph $G$ on $n$ labeled vertices. They conjectured a version of the sum-product theorem for general graphs that have at least $n^{1+\epsilon}$ edges. In this work, we consider sum-product theorems for sparse graphs, and show that this problem has important consequences already when $G$ is a matching (i.e., $n/2$ disjoint edges): Any lower bound of the form $n^{1/2+\delta}$ for its sum-product over the integers implies a lower bound of $n^{1+\delta}$ for the original Erd\H{o}s-Szemer\'edi problem. In contrast, over the reals the minimal sum-product for the matching is $\Theta(\sqrt{n})$, hence this approach has the potential of achieving lower bounds specialized to the integers. We proceed to give lower and upper bounds for this problem in different settings. In addition, we provide tight bounds for sums along expanders. A key element in our proofs is a reduction from the sum-product of a matching to the maximum number of translates of a set of integers into the perfect squares. This problem was originally studied by Euler, and we obtain a stronger form of Euler's result using elliptic curve analysis.