On addition of two distinct sets of integers

Lev, Vsevolod F.; Smeliansky, Pavel Y.

doi:10.4064/aa-70-1-85-91

Cited by 67 publications

(68 citation statements)

References 2 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In the case of the sum of two sets, Freȋman's theorem was generalized by Lev and Smeliansky; a version of it states (see [14,Lemma 1] or [17]): Going back to the case of Z/pZ, Freȋman proved [4], using trigonometric sums, that for a subset A ⊂ Z/pZ, |A+A| ≤ 2.4|A| and |A| ≤ p/35 hold only if A is contained in a short arithmetic progression. Bilu, Lev and Ruzsa [1] show that, for |A| small enough, |A+A| ≤ 3|A|−4 holds only if A is a subset of a short arithmetic progression.…”

mentioning

confidence: 99%

On the critical pair theory in Z/pZ

2006

View full text Add to dashboard Cite

Let A, B be subsets of Z/pZ such thatWe prove that, if |A| ≥ 4, |B| ≥ 5, |A + B| ≤ p − 5 and p ≥ 53 then A and B are included in arithmetic progressions with the same difference and of size |A| + 2 and |B| + 2 respectively. This extends the well-known theorem of Vosper and a recent result of Rødseth and one of the present authors.The Cauchy-Davenport Theorem [2,4] states that if A and B are subsets of Z/pZ then |A + B| ≥ min(p, |A| + |B| − 1). Vosper's Theorem [21] solves the related critical pair problem and states that, if |A|, |B| ≥ 2, then |A + B| ≥ min(p − 1, |A| + |B|)

show abstract

mentioning

confidence: 99%

On the critical pair theory in Z/pZ

2006

View full text Add to dashboard Cite

show abstract

“…The next lemma is due to Lev and Smeliansky (see [9] and [11, p.118]). Note that Lemma 1.7 can be easily modified to the following form: Let A and B both be finite subsets of {dn : n ∈ N}.…”

Section: Lemma 15 (G Freiman) Letmentioning

confidence: 99%

Inverse problem for upper asymptotic density II

Jin¹

2017

Nonstandard Methods and Applications in Mathematics

View full text Add to dashboard Cite

Inverse problems study the structure of a set A when the "size" of A + A is small. In the article, the structure of an infinite set A of natural numbers with positive upper asymptotic density is characterized when A is not a subset of an infinite arithmetic progression of difference greater than one and A + A has the least possible upper asymptotic density. For example, if the upper asymptotic density α of A is strictly between 0 and 1 2 , the upper asymptotic density of A + A is equal to 3 2 α, and A is not a subset of an infinite arithmetic progression of difference greater than one, then A is either a large subset of the union of two infinite arithmetic progressions with the same common difference k =

show abstract

“…In this section we start proving Theorem 1.1 with similar methods that Lev and Smeliansky used in [LS95] to prove (a generalisation of) Freiman's 3k−3 Theorem. More precisely, their first step was to wrap the set S modulo q := (max S − min S) and consider a subset of a finite group instead.…”

Section: Wrapping Argumentmentioning

confidence: 99%

A structure theorem for sets of small popular doubling

Mazur¹

2015

Acta Arith.

View full text Add to dashboard Cite

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.

show abstract

On addition of two distinct sets of integers

Abstract: What is the structure of a pair of finite integers sets A, B ⊂ Z with the small value of |A + B|? We answer this question for addition coefficient 3. The obtained theorem sharpens the corresponding results of G. Freiman.

Cited by 67 publications

References 2 publications

On the critical pair theory in Z/pZ

On the critical pair theory in Z/pZ

Inverse problem for upper asymptotic density II

A structure theorem for sets of small popular doubling

Contact Info

Product

Resources

About