Integral points on convex curves

The Ramanujan Journal - We estimate the maximal number of integral points which can be on a convex arc in $${mathbb {R}}^2$$ with given length, minimal radius of curvature and initial slope.     

On the Rate of Decay of the Concentration Function of the Sum of Independent Random Variables

Let X₁, ... , X_n be i.i.d. integral valued random variables and S_n their sum. In the case when X₁ has a moderately large tail of distribution, Deshouillers, Freiman and Yudin gave a uniform upper bound for max _{k ∊ ℤ} Pr{S_n = k} (which can be expressed in term of the Lévy Doeblin concentration of S_n), under the extra condition that X₁ is not essentially supported by an arithmetic progression. The first aim of the paper is to show that this extra condition cannot be simply ruled out. Secondly, it is shown that if X₁ has a very large tail (larger than a Cauchy-type distribution), then the extra arithmetic condition is not sufficient to guarantee a uniform upper bound for the decay of the concentration of the sum S_n. Proofs are constructive and enhance the connection between additive number theory and probability theory.À Jean-Louis Nicolas, avec amitié et respect2000 Mathematics Subject Classification: Primary—60Fxx, 60Exx, 11Pxx, 11B25     

Additive properties of sequences of pseudo <Emphasis Type="Italic">s</Emphasis>-th powers

In this paper, we study (random) sequences of pseudo s-th powers, as introduced by Erd?s and Rényi (Acta Arith 6:83–110, 1960). Goguel (J Reine Angew Math 278/279:63–77, 1975) proved that such a sequence is almost surely not an asymptotic basis of order s. Our first result asserts that it is however almost surely a basis of order \(s+\epsilon \) for any \(\epsilon >0\). We then study the s-fold sumset \(sA=A+\cdots +A\) (s times) and in particular the minimal size of an additive complement, that is a set B such that \(sA+B\) contains all large enough integers. With respect to this problem, we prove quite precise theorems which are tantamount to asserting that a threshold phenomenon occurs.     

A Sidon basis

We construct a Sidon set which is an asymptotic additive basis of order at most 7.