Appendix to 'Roth's theorem on progressions revisited' by J Bourgain
Classical Analysis and ODEs
2010-11-02 v4 Number Theory
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.
Cite
@article{arxiv.0710.0642,
title = {Appendix to 'Roth's theorem on progressions revisited' by J Bourgain},
author = {Tom Sanders},
journal= {arXiv preprint arXiv:0710.0642},
year = {2010}
}
Comments
11 pp. Corrected typos