SUM/DIFFERENCE/PRODUCT SETS AND GEOMETRY

by Gyorgy Elekes

Abstract:

We consider two problems of Erdo"s, both from Number Theory:

(a) Let there be given N real numbers. Consider the pairwise sums and also the pairwise products formed using these numbers. Show that at least one of these sets is large, i.e. of size close to quadratic, as a function of N.


(b) Consider a sequence of N reals where the consecutive differences strictly increase (the sequence is "convex"). Show that the set of pairwise differences formed using the N numbers is close to quadratic, as a function of N.
The lower bounds known for either case are only slightly higher than linear (!). We propose a method using Combinatorial Geometry which provides much better bounds (with very simple proofs), together with some generalizations.

(Includes joint work with I.Z.Ruzsa and M.B.Nathanson)
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