On the number of prime factors of a finite arithmetical progression
On the Product of Consecutive Elements of an Arithmetic Progression.
On the representation of integers as sums of distinct terms from a fixed set
On the structure of sets with many k-term arithmetic progressions
On the subsequence of primes having prime subscripts.
Parallelepipeds, nilpotent groups and Gowers norms
In his proof of Szemerédi’s Theorem, Gowers introduced certain norms that are defined on a parallelepiped structure. A natural question is on which sets a parallelepiped structure (and thus a Gowers norm) can be defined. We focus on dimensions and and show when this possible, and describe a correspondence between the parallelepiped structures and nilpotent groups.
Perfect powers in arithmetical progression (II)
Perfect powers in products of terms in an arithmetical progression III
Perfect powers in products of terms in an arithmetical progression (II)
Permutations avoiding arithmetic patterns.
Plus grand facteur premier d'entiers en progression arithmétique
Příspěvek k řadám arithetickým
Problème sur les progressions arithmétiques.
Product sets cannot contain long arithmetic progressions
Let B be a set of complex numbers of size n. We prove that the length of the longest arithmetic progression contained in the product set B.B = bb’ | b,b’ ∈ B cannot be greater than O((nlog²n)/(loglogn)) and present an example of a product set containing an arithmetic progression of length Ω(nlogn). For sets of complex numbers we obtain the upper bound .
Progressions arithmétiques dans les nombres premiers
Récemment, B. Green et T. Tao ont montré que : l’ensemble des nombres premiers contient des progressions arithmétiques de toutes longueurs répondant ainsi à une question ancienne à la formulation particulièrement simple. La démonstration n’utilise aucune des méthodes “transcendantes” ni aucun des grands théorèmes de la théorie analytique des nombres. Elle est écrite dans un esprit proche de celui de la théorie ergodique, en particulier de celui de la preuve par Furstenberg du théorème de Szemerédi,...
Rainbow 3-term arithmetic progressions.
Regular coverings of the integers by arithmetic progressions
Representation of odd integers as the sum of one prime, two squares of primes and powers of 2
Set-polynomials and polynomial extension of the Hales-Jewett theorem.
Sets of integers that do not contain long arithmetic progressions.