Freiman's inverse problem with small doubling property

Abstract

Let N be the set of all nonnegative integers, A ⊆ N be a finite set, and 2 A 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, Transl. Math. Monogr., vol. 37, American Mathematical Society, Providence, RI, 1973 (translated from the Russian)] the arithmetic structure of A was optimally characterized when | 2 A | ⩽ 3 | A | − 3 . 2 In [G.A. Freiman, Foundations of a Structural Theory of Set Addition, Transl. Math. Monogr., vol. 37, American Mathematical Society, Providence, RI, 1973 (translated from the Russian)] the structure of A was also characterized without proof when | 2 A | = 3 | A | − 2 . Since then the efforts have been made to generalize these results, see [V. Lev, P.Y. Smeliansky, On addition of two distinct sets of integers, Acta Arith. 70 (1) (1995) 85–91; V. Lev, On the structure of sets of integers with small doubling property, unpublished manuscripts, 1995; Y.O. Hamidoune, A. Plagne, A generalization of Freiman's 3 k − 3 theorem, Acta Arith. 103 (2) (2002) 147–156] for example. However, no optimal characterization of the structure of A has been obtained without imposing extra conditions, until now, when | 2 A | > 3 | A | − 2 . In this paper we optimally characterize, with the help of nonstandard analysis, the arithmetic structure of A when | 2 A | = 3 | A | − 3 + b , where b is positive but not too large. Precisely, we prove that there is a real number ϵ > 0 and there is K ∈ N such that if | A | > K and | 2 A | = 3 | A | − 3 + b for 0 ⩽ b ⩽ ϵ | A | , then A is either a subset of an arithmetic progression of length at most 2 | A | − 1 + 2 b or a subset of a bi-arithmetic progression 3 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 [R. Jin, Solution to the inverse problem for upper asymptotic density, J. Reine Angew. Math. (Crelle's J.) 595 (2006) 121–166]. 2 One can easily prove that | 2 A | ⩾ 2 | A | − 1 is always true and | 2 A | = 2 | A | − 1 implies that A is an arithmetic progression. 3 The definition of bi-arithmetic progression can be found in the beginning of Introduction.