2009/04/04 by Ernie Croot, Croot, Ernie, Derrick Hart +1
Computer Science · Mathematics · #Advanced Graph Theory Research #Graph Labeling and Dimension Problems #Limits and Structures in Graph Theory #math.CO #math.NT #msc:11B99
paper · pdf · doi:10.48550/arxiv.0904.0718
19 pages. Final draft -- light corrections, submitted to Siam J. of Discrete Math
arxiv created 2009/04/15 · arxiv updated 2009/12/01
In the present paper we show that if A is a set of n real numbers, and the product set A.A has at most n^(1+c) elements, then the k-fold sumset kA has at least n^(log(k/2)/2 log 2 + 1/2 - fk(c)) elements, where fk(c) -> 0 as c -> 0. We believe that the methods in this paper might lead to a much stronger result; indeed, using a result of Trevor Wooley on Vinogradov's Mean Value Theorem and the Tarry-Escott Problem, we show that if |A.A| < n^(1+c), then |k(A.A)| > n^(Omega((k/log k)^(1/3))), for c small enough in terms of k (we believe that a certain modification of this argument can perhaps produce similar conclusions for kA).