Multiple recurrence and nilsequences

Aiming at a simultaneous extension of Khintchine(X,X,m,T)(X,\mathcal{X},\mu,T) and a set A ? XA\in\mathcal{X} of positive measure, the set of integers n such that A T^2nA T^knA)(A)^k+1-\mu(A{\cap} T^{n}A{\cap} T^{2n}A{\cap} \ldots{\cap} T^{kn}A)>\mu(A)^{k+1}-\epsilon is syndetic. The size of this set, surprisingly enough, depends on the length (k+1) of the arithmetic progression under consideration. In an ergodic system, for k=2 and k=3, this set is syndetic, while for kòf(x)f(Tⁿx)f(T²ⁿx)? f(T^knx)  dm(x)\int{f(x)f(T^{n}x)f(T^{2n}x){\ldots} f(T^{kn}x) \,d\mu(x)} , where k and n are positive integers and f is a bounded measurable function. We also derive combinatorial consequences of these results, for example showing that for a set of integers E with upper Banach density d^*(E)>0 and for all {n ? \mathbbZ\colon d^*(E?(E+n)?(E+2n)?(E+3n)) > d^*(E)⁴-e}\big\{n\in\mathbb{Z}{\colon} d^*\big(E\cap(E+n)\cap(E+2n)\cap(E+3n)\big) > d^*(E)^4-\epsilon\big\}     

On the small sieve. I. Sifting by primes

The main object of the paper is to prove that if P is a set of primes with sum of reciprocals ≤K, then the number of natural numbers up to x, divisible by no element of P, is ≥cx, where c is a positive constant depending only on K. A lower estimate is given for c and a similar result is achieved in the case when the condition of primality is substituted by the weaker condition that any m elements of the sifting set are coprime.     

Sum-Avoiding Subsets

We estimate the maximal number of elements that can be selected from a set of n numbers so that the sum of two selected elements are never in the original set. We improve Choi’s upper bound of n^2/5 to e^{c√log n n}.To Professor Nicolas, on the occasion of his 60th birthday.2000 Mathematics Subject Classification: Primary—11B75Supported by Hungarian National Foundation for Scientific Research (OTKA), Grants No. T 38396, T43623 and T42750.     

On the Exponential sum over r-Free Integers

We determine, up to a constant factor, the L ¹ mean of the exponential sum formed with the r-free integers. This improves earlier results of Brüdern, Granville, Perelli, Vaughan and Wooley. As an application, we improve the known bound for the L ¹ norm of the exponential sum defined with the Möbius function.     

The structure of sets with few sums along a graph

We present common generalizations of some structure results of Freiman, Ruzsa, Balog-Szemerédi and Laczkovich-Ruzsa. We also give some applications to Combinatorial Geometry and Algebra, some of which generalize the aforementioned structure results even further.     

Additive completion and disjoint translations

Supported by Hungarian National Foundation for Scientific Research, Grant No. 1901.     

A just basis

An old problem of P. Erdös and P. Turán asks whether there is a basisA of order 2 for which the number of representationsn=a+a′, a,a′∈A is bounded. Erd?s conjectured that such a basis does not exist. We answer a related finite problem and find a basis for which the number of representations is bounded in the square mean. Writing σ (n)=|{(a, a ^t ) ∈A ²:a+a′=n}| we prove that there exists a setA of nonnegative integers that forms a basis of order 2 (that is,s(n)≥1 for alln), and satisfies ∑_{n ? N} σ(N)² = O(N).     

Ein neues formales System der deontischen Logik

Ohne Zusammenfassung Vorgelegt von L. Kalmár