Nil-Bohr Sets of Integers

Bernard Host (1) , Bryna Kra (2)

1 LAMA - Laboratoire d'Analyse et de Mathématiques Appliquées
2 Department of Mathematics, Northwestern University

We study relations between subsets of integers that are large, where large can be interpreted in terms of size (such as a set of positive upper density or a set with bounded gaps) or in terms of additive structure (such as a Bohr set). Bohr sets are fundamentally abelian in nature and are linked to Fourier analysis. Recently it has become apparent that a higher order, non-abelian, Fourier analysis plays a role in both additive combinatorics and in ergodic theory. Here we introduce a higher order version of Bohr sets and give various properties of these objects, generalizing results of Bergelson, Furstenberg, and Weiss.

Mots clés: Bohr sets; nilsystems; sumsets

Domaines: Systèmes dynamiques [math.DS]

Bernard Host, Bryna Kra. Nil-Bohr Sets of Integers. Ergodic Theory and Dynamical Systems, 2011, 31 (1), pp.113-142. ⟨hal-00366847⟩