Freiman’s inverse problem with small doubling property. (English) Zbl 1231.11012

Jin, Renling

Adv. Math. 216, No. 2, 711-752 (2007).

Summary: Let \(\mathbb N\) be the set of all non-negative integers, \(A\subseteq\mathbb N\) be a finite set, and \(2A\) be the set of all numbers of the form \(a+b\) for all \(a\) and \(b\) in \(A\). In [G. A. Freiman, Foundations of a structural theory of set addition. Translated from the Russian. Translations of Mathematical Monographs, Vol 37. Providence, R. I.: AMS (1973; Zbl 0271.10044)] the arithmetic structure of \(A\) was optimally characterized when \(|2A|\leq 3|A| -3\). In the same monograph the structure of \(A\) was also characterized without proof when \(|2A| = 3|A|-2\). Since then several efforts have been made to generalize these results. However, no optimal characterization of the structure of \(A\) has been obtained without imposing extra conditions, until now, when \(|2A| > 3|A| - 2\).
In this paper we optimally characterize, with the help of nonstandard analysis, the arithmetic structure of \(A\) when \(|2A| = 3|A| - 3 + b\), where \(b\) is positive but not too large. Precisely, we prove that there is a real number \(\varepsilon > 0\) and there is \(K\in\mathbb N\) such that if \(|A| > K\) and \(|2A| = 3|A|-3+b\) for \(0\leq b \leq \varepsilon |A|\), then \(A\) is either a subset of an arithmetic progression of length at most \(2|A|-1+2b\) or a subset of a bi-arithmetic progression of length at most \(|A|+b\). An application of this result to the inverse problem for upper asymptotic density is presented near the end of the paper. In the application we improve the most important part of the main theorem in the author’s article in [J. Reine Angew. Math. 595, 121–165 (2006; Zbl 1138.11045)].

MSC:

11B05	Density, gaps, topology
11P70	Inverse problems of additive number theory, including sumsets
11B13	Additive bases, including sumsets
11U10	Nonstandard arithmetic (number-theoretic aspects)
03H15	Nonstandard models of arithmetic

Keywords:

inverse problem; Freiman’s theorem; additive number theory; nonstandard analysis

Citations:

Zbl 0271.10044; Zbl 1138.11045

References:

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