Combinatorial properties of Nil-Bohr sets

Abstract

In this paper we study the relation between two notions of largeness that apply to a set of positive integers, namely ${\mathrm{Nil}}_d-\mathrm{Bohr}$ and ${\mathrm{SG}}_k$, as introduced by Host and Kra. We prove that any ${\mathrm{Nil}}_d-{\mathrm{Bohr}}_0$ set is necessarily ${\mathrm{SG}}_k$ where $k$ is effectively bounded in terms of $d$. This partially resolves a conjecture of Host and Kra.

25 pages

References in corpus (3)

• A nilpotent IP polynomial multiple recurrence theorem
• Ergodic theorems for polynomials in nilpotent groups
• IPr* recurrence and nilsystems