## Appendix to ‘Roth’s theorem on progressions revisited’ by J. Bourgain.(English)Zbl 1222.11014

This paper contains three theorems, two of which are stated without proof in J. Bourgain’s paper [J. Anal. Math. 104, 155–192 (2008; Zbl 1155.11011)]. The authorship of these results is rather obscure to the reviewer.
The third theorem, which is logically the first, asserts that for two sets $$A, B \subset {\mathbb {Z}}/N{\mathbb {Z}}$$ such that $$| A+B| \leq K| B|$$ and $$| B| =\alpha N$$, the second difference set $$2A-2B$$ contains a rather large Bohr set. This is applied to improve the estimate in Freiman’s theorem asserting that a set $$A\subset {\mathbb {Z}}$$ satisfying $$| 2Aw\leq K| A|$$ can be covered by a multidimensional arithmetic progression. The new estimate for the dimension is $$O(K^{7/4+\varepsilon })$$ and for the size $$| A| \exp K^{7/4+\varepsilon }$$.
The third asserts that for a finite set $$A\subset {\mathbb {R}}$$ and a transcendental $$\alpha$$ we have $| A+ \alpha \cdot A| \gg | A| ( \log | A| )^{4/3} ( \log \log | A| )^{-8/3} .$ This improves a result of Konyagin and Łaba, where the exponent of the logarithm was 2. As an upper estimate we have $$| A| \exp O(\sqrt { \log | A| })$$; the reviewer thinks the truth will be near this.

### MSC:

 11B25 Arithmetic progressions 11P55 Applications of the Hardy-Littlewood method

### Keywords:

arithmetic progression; Bohr set; sumsets

Zbl 1155.11011
Full Text:

### References:

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