An elementary proof of Jin's theorem with a bound

We present a short and self-contained proof of Jin's theorem about the piecewise syndeticity of dierence sets which is entirely elementary, in the sense that no use is made of nonstandard analysis, ergodic theory, measure theory, ultralters, or other advanced tools. An explicit bound to the number of shifts that are needed to cover a thick set is provided. Precisely, we prove the following: If A and B are sets of integers having positive upper Banach densities a and b respectively, then there exists a finite set F of cardinality at most 1/ab such that (A-B) + F covers arbitrarily long intervals

Extremal densities and measures on groups and GG-spaces and their combinatorial applications

This text contains lecture notes of the course taught to Ph.D. students of Jagiellonian University in Krakow on 25-28 November, 2013.Comment: 18 page

Characterizing the structure of A+B when A+B has small upper Banach density

AbstractLet A and B be two infinite sets of non-negative integers. Similar to Kneser's Theorem (Theorem 1.1 below) we characterize the structure of A+B when the upper Banach density of A+B is less than the sum of the upper Banach density of A and the upper Banach density of B

The Solecki submeasures and densities on groups

We introduce the Solecki submeasure $\sigma(A)=\inf_F\sup_{x,y\in G}|F\cap xAy|/|F|$ and its left and right modifications on a group $G$, and study the interplay between the Solecki submeasure and the Haar measure on compact topological groups. Also we show that the right Solecki density on a countable amenable group coincides with the upper Banach density $d^*$ which allows us to generalize some fundamental results of Bogoliuboff, Folner, Cotlar and Ricabarra, Ellis and Keynes about difference sets and Jin, Beiglbock, Bergelson and Fish about the sumsets to the class of all amenable groups.Comment: 34 page

Nonstandard Methods in Ramsey Theory and Combinatorial Number Theory

The goal of this present manuscript is to introduce the reader to the nonstandard method and to provide an overview of its most prominent applications in Ramsey theory and combinatorial number theory.Comment: 126 pages. Comments welcom