A structure theorem for sets of small popular doubling

Abstract

Abstract. In this paper we prove that every set A ⊂ Z satisfying the inequality x min(1 A * 1 A (x), t) (2 + δ)t|A| for t and δ in suitable ranges, then A must be very close to an arithmetic progression. We use this result to improve the estimates of Green and Morris for the probability that a random subset A ⊂ N satisfies |N \ (A + A)| k; specifically we show that P(|N \ (A + A)| k) = Θ(2 −k/2 ). IntroductionLet us start with recalling Freiman (3k − 3) Theorem. It states that every finite subset A ⊂ Z satisfying |A + A| < 3|A| − 3 is contained in an arithmetic progression of length |A + A| − |A| + 1. Comparing this with a lower bound |A + A| 2|A| − 1 valid for all nonempty finite subsets of Z, we can see that this result describes sets for quite large range of values of |A + A|. Our goal is to give a similar result for a set with a few popular sums. Note that it cannot be done dirctly; the reason is that the set S k (A) = {x ∈ Z : |A ∩ (x − A)| k} of k-popular sums is empty if k 3 and A is a highly independent set. Instead, we need to consider a different quantity, namely the average size of S k for 1 k t, which also appeared quite natural to Pollard in his work [Pol74] back in 1974.At this point it is convenient to use the notation of convolution. From now on, we will consider any abelian group G to be equipped with the couning measure, which leads to the definitionfor any functions f, g : G → C for which the above expression makes sense (i.e. is absolutely convergent; we will use it mostly for f, g being indicator functions of finite sets). Having this notation, we can restate Pollard's theorem asfor any prime p and sets A, B ⊂ Z/pZ. It is not hard to prove the corresponding statement for subsets of the integers (and even easier to deduce it from Pollard's theorem); in particular, for a single set A ⊂ Z and an integer 2010 Mathematics Subject Classification. 11P70.

On the number of sets with a given doubling constant

“…To prove a stability theorem for almost all sets with a given size and doubling constant we will also need the following result of Mazur [10]. From Theorem 3.4 we can easily deduce the following corollary:…”

“…ǫKs, such10 that B \ T ⊂ P . Observe that, there at most s ′ ≥2 9 ǫs ⊂ B that are not (2 4 ǫ, Ks)-close to an arithmetic progression, since they must have s − s ′ elements in B \ T and s ′ elements in T for some s ′ ≥ 2 9 ǫs.…”

A robust version of Freiman's 3k–4 Theorem and applications

“…In Section 3, we develop robust analgoues of Kneser's theorem in abelian groups, which are used in Section 4 to deduce the technical propositions. In Section 5, we relate our main result with similar types of results in the literature [10][11][12]. Section 6 contains the deduction of Theorem 1•3 from Theorem 1•1, and finally in Section 7 we deduce a continuous version of Theorem 1•1, which is a special case of previous works on near equality in the Riesz-Sobolev inequality [1][2][3][4].…”

Untitled

“…We would prefer to state a specific δ in terms of ε in Theorems 1.1 and 1.15, but our present methods will not yield any quantitative dependence. With additional hypotheses on the ambient group G, and in some cases on m(A) and m(B), such quantitative results are provided in [3,8,7,11,32,33,40], and Chapters 19 and 21 of [24]. A prototypical example of such a result is [34,Theorem 2.11] (originally in [13]).…”

“…Theorem 1.1 generalizes Theorem 1.5 of [44], which imposes the additional assumption that G is connected. A quantitative version of this result in the case where G is a cyclic group of prime order is proved in [32]. See also [33] for a version in Z, and more recently [41].…”