k-fold sums from a set with few products

TL;DR

This paper investigates the relationship between the size of the product set and the k-fold sumset of a real number set, establishing lower bounds under small product set conditions and suggesting potential for stronger results using advanced number theory techniques.

Contribution

It introduces new lower bounds for the size of k-fold sumsets when the product set is small, and proposes methods that could lead to significantly stronger results.

Findings

Lower bounds for |kA| based on small |A.A|

Application of Vinogradov's Mean Value Theorem

Potential for improved bounds using advanced techniques

Abstract

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 - f_k(c)) elements, where f_k(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).