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A polynomial bound in Freiman’s theorem. (English) Zbl 1035.11048
In the fifties, Freiman started to study the sets of integers with the small doubling property. His famous theorem states that, for $$\alpha$$ a real number, the sets of integers $$A$$ such that $$| A+ A|\leq\alpha| A|$$ are “structured” in the sense that $$A$$ has to be contained in a $$d$$-dimensional arithmetic progression, that is a set of the form $P= \Biggl\{p_0+ \sum^d_{i=1} x_i p_i\text{ where }0\leq x_i\leq d_i\Biggr\},$ that $$A$$ fulfills well (by which we mean $$| A|\geq c| P|$$ for some $$c > 0$$). The point is that $$d$$ and $$c$$ are bounded by a function depending only on $$\alpha$$. This theorem is included in G. A. Freiman’s famous book [Foundations of a structural theory of set addition (Russian), Kazan. Gos. Ped. Inst., Elabuzh. Gos. Ped. Inst., Kazan (1966; Zbl 0203.35305)] and its English translation [Translations of Mathematical Monographs, Vol. 37, AMS (1973; Zbl 0271.10044)].
There has been some progress (and new proofs) on Freiman’s original result: see I. Z. Ruzsa [Acta Math. Hung. 65, 379–388 (1994; Zbl 0816.11008)] and Y. Bilu [Structure of sets with small sumset. Structure theory of set addition. Astérisque 258, 77–108 (1999; Zbl 0946.11004)]. In particular, effective estimates on the functions $$d(\alpha)$$ and $$c(\alpha)$$ were established.
In the present paper, improved estimates are proved, namely $$d(\alpha)\leq [\alpha- 1]$$ and $$c(\alpha)\geq\exp(-c_0 \alpha^2\log^3\alpha)$$ (for some constant $$c_0$$). The bound on $$c(\alpha$$) is a drastic improvement since earlier bounds provided doubly exponential estimates.
The proof of this important result combines the – locally improved – known proofs of Freiman’s theorem (mainly that one of Ruzsa) and new harmonic analysis arguments.

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