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Arithmetic progressions in sumsets

Imre Z. Ruzsa — 1991

Acta Arithmetica

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 way. Observe...

Sumsets of Sidon sets

Imre Z. Ruzsa — 1996

Acta Arithmetica

1. Introduction. A Sidon set is a set A of integers with the property that all the sums a+b, a,b∈ A, a≤b are distinct. A Sidon set A⊂ [1,N] can have as many as (1+o(1))√N elements, hence  N/2 sums. The distribution of these sums is far from arbitrary. Erdős, Sárközy and T. Sós [1,2] established several properties of these sumsets. Among other things, in [2] they prove that A + A cannot contain an interval longer than C√N, and give an example that N 1 / 3 is possible. In [1] they show that A + A contains...

Polynomial growth of sumsets in abelian semigroups

Melvyn B. NathansonImre Z. Ruzsa — 2002

Journal de théorie des nombres de Bordeaux

Let S be an abelian semigroup, and A a finite subset of S . The sumset h A consists of all sums of h elements of A , with repetitions allowed. Let | h A | denote the cardinality of h A . Elementary lattice point arguments are used to prove that an arbitrary abelian semigroup has polynomial growth, that is, there exists a polynomial p ( t ) such that | h A | = p ( h ) for all sufficiently large h . Lattice point counting is also used to prove that sumsets of the form h 1 A 1 + + h r A r have multivariate polynomial growth.

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