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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

Citations:

Zbl 1155.11011
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References:

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