# Cyclicity in weighted $\ell^p$ spaces

## TL;DR

This paper investigates the conditions under which sequences in weighted ^p spaces are cyclic, linking these conditions to the Hausdorff dimension and capacity of the Fourier transform's zero set.

## Contribution

It provides new necessary and sufficient conditions for cyclicity in weighted ^p spaces based on geometric properties of the Fourier transform's zero set.

## Key findings

- Characterizes cyclicity using Hausdorff dimension.
- Links capacity of zero set to cyclicity.
- Establishes criteria for dense span in weighted ^p spaces.

## Abstract

We study the cyclicity in weighted $\ell^p(\mathbb{Z})$ spaces. For $p \geq 1$ and $\beta \geq 0$, let $\ell^p\_\beta(\mathbb{Z})$ be the space of sequences $u=(u\_n)\_{n\in \mathbb{Z}}$ such that $(u\_n |n|^{\beta})\in \ell^p(\mathbb{Z}) $. We obtain both necessary conditions and sufficient conditions for $u$ to be cyclic in $\ell^p\_\beta(\mathbb{Z})$, in other words, for $ \{(u\_{n+k})\_{n \in \mathbb{Z}},~ k \in \mathbb{Z} \}$ to span a dense subspace of $\ell^p\_\beta(\mathbb{Z})$. The conditions are given in terms of the Hausdorff dimension and the capacity of the zero set of the Fourier transform of $u$.