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Near optimal bounds in Freiman’s theorem. (English) Zbl 1242.11074

The author shows that there is an absolute constant \(C>0\) such that each finite set \(A\) of integers satisfying \(|A+A|\leq K|A|\) is contained in a generalized arithmetic progression of dimension at most \(d(K)\) and size at most \(f(K)|A|\), with \(d(k)\leq K^{1+C/\sqrt{\log K}}\) and \(f(K)\leq \text{exp}(K^{1+C/\sqrt{\log K}})\). This improves previous quantitative versions of Freiman’s theorem. The proof involves a refinement of the Bogolyubov-Ruzsa lemma. The author mentions that his method can also be applied to yield a quantitative version of Freiman’s theorem in the “torsion” case.

MSC:

11P70 Inverse problems of additive number theory, including sumsets
11B13 Additive bases, including sumsets
11B25 Arithmetic progressions
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References:

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