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

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

2007-10-02

arxiv.org/abs/0710.0642v4

J. Anal. Math. 104 (2008), 193-206

Mathematics

Classical Analysis and ODEs

11 pp. Corrected typos

Scientific paper

10.1007/s11854-008-0021-9

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.

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