Displaying similar documents to “Large sets with small doubling modulo p are well covered by an arithmetic progression”

Arithmetic progressions in sumsets

Imre Z. Ruzsa (1991)

Acta Arithmetica

Similarity:

1. Introduction. Let A,B ⊂ [1,N] be sets of integers, |A|=|B|=cN. Bourgain [2] proved that A+B always contains an arithmetic progression of length e x p ( l o g N ) 1 / 3 - ε . Our aim is to show that this is not very far from the best possible. Theorem 1. Let ε be a positive number. For every prime p > p₀(ε) there is a symmetric set A of residues mod p such that |A| > (1/2-ε)p and A + A contains no arithmetic progression of length (1.1) e x p ( l o g p ) 2 / 3 + ε . A set of residues can be used to get a set of integers in an obvious...

A canonical Ramsey-type theorem for finite subsets of ℕ

Diana Piguetová (2003)

Commentationes Mathematicae Universitatis Carolinae

Similarity:

T. Brown proved that whenever we color 𝒫 f ( ℕ ) (the set of finite subsets of natural numbers) with finitely many colors, we find a monochromatic structure, called an arithmetic copy of an ω -forest. In this paper we show a canonical extension of this theorem; i.eẇhenever we color 𝒫 f ( ℕ ) with arbitrarily many colors, we find a canonically colored arithmetic copy of an ω -forest. The five types of the canonical coloring are determined. This solves a problem of T. Brown.

On a problem of Matkowski

Zoltán Daróczy, Gyula Maksa (1999)

Colloquium Mathematicae

Similarity:

We solve Matkowski's problem for strictly comparable quasi-arithmetic means.

Kneser’s theorem for upper Banach density

Prerna Bihani, Renling Jin (2006)

Journal de Théorie des Nombres de Bordeaux

Similarity:

Suppose A is a set of non-negative integers with upper Banach density α (see definition below) and the upper Banach density of A + A is less than 2 α . We characterize the structure of A + A by showing the following: There is a positive integer g and a set W , which is the union of ⌈ 2 α g - 1 ⌉ arithmetic sequences [We call a set of the form a + d ℕ an arithmetic sequence of difference d and call a set of the form { a , a + d , a + 2 d , ... , a + k d } an arithmetic progression of difference d . So an arithmetic progression is finite and an arithmetic...

Some additive applications of the isoperimetric approach

Yahya O. Hamidoune (2008)

Annales de l’institut Fourier

Similarity:

Let G be a group and let X be a finite subset. The isoperimetric method investigates the objective function | ( X B ) ∖ X | , defined on the subsets X with | X | ≥ k and | G ∖ ( X B ) | ≥ k , where X B is the product of X by B . In this paper we present all the basic facts about the isoperimetric method. We improve some of our previous results and obtain generalizations and short proofs for several known results. We also give some new applications. Some of the results obtained here will be used in coming...