Strongly central sets and sets of polynomial returns mod 1

Abstract

Central sets in <!-- MATH $\mathbb{N}$ --> were introduced by Furstenberg and are known to have substantial combinatorial structure. For example, any central set contains arbitrarily long arithmetic progressions, all finite sums of distinct terms of an infinite sequence, and solutions to all partition regular systems of homogeneous linear equations. We introduce here the notions of strongly central and very strongly central, which as the names suggest are strictly stronger than the notion of central. They are also strictly stronger than syndetic, which in the case of <!-- MATH $\mathbb{N}$ --> means that gaps are bounded.