We show that for every positive integer k there are positive constants C and c such that if A is a subset of {1, 2, …, n} of size at least C n1/k, then, for some d ≤ k-1, the set of subset sums of A contains a homogeneous d-dimensional generalized arithmetic progression of size at least c|A|d+1. This strengthens a result of Szemerédi and Vu, who proved a similar statement without the homogeneity condition. As an application, we make progress on the Erdős-Straus non-averaging sets problem, showing that every subset A of {1, 2, …, n} of size at least n√2 – 1 + o(1) contains an element which is the average of two or more other elements of A. This gives the first polynomial improvement on a result of Erdős and Sárközy from 1990.
This video is part of the Institute for Advanced Study‘s Special year research seminar.
