Sums and products along sparse graphs

Noga Alon, Omer Angel, Itai Benjamini, Eyal Lubetzky

Research output: Contribution to journalArticle

9 Scopus citations

Abstract

In their seminal paper from 1983, Erdo{double acute}s and Szemerédi showed that any n distinct integers induce either n1+e{open} distinct sums of pairs or that many distinct products, and conjectured a lower bound of n2-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 n1+e{open} 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 n1/2+δ for its sum-product over the integers implies a lower bound of n1+δ for the original Erdo{double acute}s-Szemerédi problem. In contrast, over the reals the minimal sum-product for the matching is {circled dash}(√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.

Original languageEnglish (US)
Pages (from-to)353-384
Number of pages32
JournalIsrael Journal of Mathematics
Volume188
Issue number1
DOIs
StatePublished - Mar 1 2012
Externally publishedYes

All Science Journal Classification (ASJC) codes

  • Mathematics(all)

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