Vectors with orthogonal iterates along IP-sets in unitary group actions

Abstract

We prove the following result in commutative harmonic analysis.

THEOREM 1. Suppose\muis a probability measure on the dual\hat Gof a countable Abelian groupG, Eis a symmetric subset ofGand\sum_{g \in E-E, g \neq 0} |\widehat{\mu} (g)| < \delta < 1/3.Then there is anf \in C^+(\hat G)such that\|f-1\| < \tfrac{3}{2}\delta, (\widehat{f\mu})(g)=0for allg\in E\backslash \{0\}and(\widehat{f\mu})(0)=1.

This has the following corollary which leads to simple proofs and improvements of many of the results in a recent paper of Bergelson et al.

THEOREM 2.Ifg \mapsto U_gis a unitary action of a discrete Abelian groupGon a Hilbert space{\cal H}and\{g_i\} \subset Gis a sequence such that both\{g_i\}and\{2g_i\}are mixing for this action then for allx\in{\cal H}and\epsilon>0there are anx'\buildrel\epsilon\over\sim xand an IP-set\Gammagenerated by a subsequence of\{g_i\}such that the vectors\{U_gx':g\in\Gamma\}are pairwise orthogonal.

We also show that even in the case of a \mathbb{Z}-action (i.e. a single unitary operator U) the hypothesis \{g_i\} mixing alone is not sufficient for the conclusion of Theorem 2. Finally we give conditions on G and on the spectrum of the action under which that assumption is sufficient.