Bohr recurrence and density of non-lacunary semigroups of ℕ

2024/06/03 by Nikos Frantzikinakis, Frantzikinakis, Nikos, Bernard Host +3
Mathematics · #advanced mathematical theories #Functional Equations Stability Results #Mathematical Dynamics and Fractals

paper · pdf · doi:10.48550/arxiv.2406.01353

Abstract

A subset R of integers is a set of Bohr recurrence if every rotation on \mathbbT^d returns arbitrarily close to zero under some non-zero multiple of R. We show that the set \k! 2^m3ⁿ\colon k,m,n∈ ℕ\ is a set of Bohr recurrence. This is a particular case of a more general statement about images of such sets under any integer polynomial with zero constant term. We also show that if P is a real polynomial with at least one non-constant irrational coefficient, then the set \P(2^m3ⁿ)\colon m,n∈ ℕ\ is dense in \mathbbT, thus providing a joint generalization of two well-known results, one of Furstenberg and one of Weyl.

Related

• Nil Bohr₀-sets, Poincaré recurrence and generalized polynomials
• Some new examples of recurrence and non-recurrence sets for products of rotations on the unit circle
• Special cases and equivalent forms of Katznelson's problem on recurrence
• 2-large sets are sets of Bohr recurrence
• On Bohr sets of integer valued traceless matrices