TY - JOUR

T1 - Sums and products along sparse graphs

AU - Alon, Noga

AU - Angel, Omer

AU - Benjamini, Itai

AU - Lubetzky, Eyal

N1 - Funding Information:
∗Research of N. Alon was supported in part by a USA Israeli BSF grant, by a grant from the Israel Science Foundation, by an ERC Advanced Grant and by the Hermann Minkowski Minerva Center for Geometry at Tel Aviv University. ∗∗ Research of O. Angel was supported by NSERC and the University of Toronto Received September 8, 2009 and in revised form September 23, 2010

PY - 2012/3

Y1 - 2012/3

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=84858451111&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84858451111&partnerID=8YFLogxK

U2 - 10.1007/s11856-011-0170-x

DO - 10.1007/s11856-011-0170-x

M3 - Article

AN - SCOPUS:84858451111

SN - 0021-2172

VL - 188

SP - 353

EP - 384

JO - Israel Journal of Mathematics

JF - Israel Journal of Mathematics

IS - 1

ER -