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### Diameter of Sets and Measure of Sumsets.

Monatshefte für Mathematik

### A Just Basis.

Monatshefte für Mathematik

### Uniform distrubution, positive trigonometric polynomials and difference sets

Seminaire de Théorie des Nombres de Bordeaux

### Logarithmic Density and Measures on Semigroups.

Manuscripta mathematica

### On sets of weak uniform distribution

Colloquium Mathematicae

Acta Arithmetica

### Arithmetic progressions in sumsets

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 $exp{\left(logN\right)}^{1/3-\epsilon }$. 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)$exp{\left(logp\right)}^{2/3+\epsilon }$. A set of residues can be used to get a set of integers in an obvious way. Observe...

Acta Arithmetica

Acta Arithmetica

### Sumsets of Sidon sets

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...

Acta Arithmetica

### Additive properties of dense subsets of sifted sequences

Journal de théorie des nombres de Bordeaux

We examine additive properties of dense subsets of sifted sequences, and in particular prove under very general assumptions that such a sequence is an additive asymptotic basis whose order is very well controlled.

### Polynomial growth of sumsets in abelian semigroups

Journal de théorie des nombres de Bordeaux

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

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