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


N. Wiener characterized the cyclic vectors (with respect to translations) in $\ell^p(\mathbb{Z})$ and $L^p(\mathbb{R})$, $p=1,2$, 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.


Nir Lev

Faculty of Mathematics and Computer Science
The Weizmann Institute of Science
Rehovot 76100

Alexander Olevskii

School of Mathematical Sciences
Tel Aviv University
Ramat Aviv
Tel Aviv, 69978