Appendix to 'Roth's theorem on progressions revisited' by J Bourgain

TL;DR

This paper refines Freiman's theorem for integer sets with small sumsets and improves bounds on sum-product estimates for real sets involving transcendental numbers, advancing additive combinatorics understanding.

Contribution

It provides a sharper structure theorem for sets with small sumsets and enhances sum-product estimates involving transcendental numbers.

Findings

A multidimensional progression containing A with controlled dimension and size.

An improved lower bound on |A + aA| for real sets with transcendental a.

Enhanced understanding of additive and multiplicative structure in finite sets.

Abstract

We show two results. First, a refinement of Freiman's theorem: if A is a finite set of integers and |A+A| < K|A|, then A is contained in a multidimensional progression of dimension at most O(K^{7/4} log^3K) and size at most exp(O(K^{7/4} log^3K))|A|. Secondly, an improvement of a result of Konyagin and Laba: if A is a finite set of reals and a is a transcendental then |A+aA| >> |A|(log |A|)^{4/3-\epsilon} for all \epsilon>0.