The independence number of a subset of an Abelian group.
General multiplicative functions
On an additive property of squares and primes
Semigroup-valued multiplicative functions
The density of the set of sums
On difference-sequences
A concavity property for the measure of product sets in groups
Let G be a connected locally compact group with a left invariant Haar measure μ. We prove that the function ξ(x) = inf μ̅(AB): μ(A) = x is concave for any fixed bounded set B ⊂ G. This is used to give a new proof of Kemperman’s inequality for unimodular G.
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
Prime values of reducible polynomials, I
An application of Kloosterman sums
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