Borys Kuca

Polynomial Szemeredi theorem

Additive combinatorics studies the presence of patterns such as arithmetic progressions in subsets of natural numbers or abelian groups. These structures can be examined using ideas from combinatorics, number theory, analysis and dynamics. One of the best known results in the area is Szemeredi theorem, which states that each dense subset of natural numbers contains an arithmetic progression of arbitrary length. Among its generalisations is a result of Bergelson and Leibman, who showed that the same holds for any polynomial progression x, x+P_1(y), …, x+P_m(y) with P_1, …, P_m having integer coefficients and zero constant terms. In this talk, I will discuss the theorem of Bergelson and Leibman together with recent developments.