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

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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

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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

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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

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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...