Diameter of Sets and Measure of Sumsets.
A Just Basis.
Uniform distrubution, positive trigonometric polynomials and difference sets
Logarithmic Density and Measures on Semigroups.
On sets of weak uniform distribution
Solving a linear equation in a set of integers I
Arithmetic progressions in sumsets
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 . 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). A set of residues can be used to get a set of integers in an obvious way. Observe...
Solving a linear equation in a set of integers II
On the additive completion of primes
Sumsets of Sidon sets
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 is possible. In [1] they show that A + A contains...
Negative values of cosine sums
An application of Kloosterman sums
Convolution Quotients of Nonnegative Functions.
Additive properties of dense subsets of sifted sequences
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
Let be an abelian semigroup, and a finite subset of . The sumset consists of all sums of elements of , with repetitions allowed. Let denote the cardinality of . Elementary lattice point arguments are used to prove that an arbitrary abelian semigroup has polynomial growth, that is, there exists a polynomial such that for all sufficiently large . Lattice point counting is also used to prove that sumsets of the form have multivariate polynomial growth.
A set of squares without arithmetic progressions
Difference sets and the primes
Prime values of reducible polynomials, II
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