Wiener's 'closure of translates' problem and Piatetski-Shapiro's uniqueness phenomenon

TL;DR

This paper investigates Wiener's problem on the characterization of cyclic vectors in function spaces, providing a counterexample to Wiener's conjecture for the range 1<p<2, and explores related uniqueness phenomena.

Contribution

The paper disproves Wiener's conjecture for 1<p<2 by constructing a counterexample, advancing understanding of cyclic vectors and Fourier transform zero sets.

Findings

Counterexample to Wiener's conjecture for 1<p<2

Disproof of the conjecture's characterization in certain function spaces

Enhanced understanding of Piatetski-Shapiro's uniqueness phenomenon

Abstract

Wiener characterized the cyclic vectors (with respect to translations) in $l^p(Z)$ and $L^p(R)$, $p=1,2$, in terms of the zero set of the Fourier transform. He conjectured that a similar characterization should be true for $1<p<2$. Our main result contradicts this conjecture.