Completeness of sparse, almost integer and finite local complexity sequences of translates in $L^p(\mathbb{R})$

Abstract

A real sequence $\hat{I}›=\{\hat{I}{»}_n{\}}_{n=1}^{\infty }$ is called $p$-generating if there exists a function $g$ whose translates $\{g(x-\hat{I}{»}_n){\}}_{n=1}^{\infty }$ span the space $L^p(\mathbb{R})$. While the $p$-generating sets were completely characterized for $p=1$ and $p>2$, the case $1<p\le 2$ remains not well understood. In this case, both the size and the arithmetic structure of the set play an important role. In the present paper, (i) We show that a $p$-generating set $\hat{I}›$ of positive real numbers can be very sparse, namely, the ratios $\hat{I}{»}_{n+1}/\hat{I}{»}_n$ may tend to $1$ arbitrarily slowly; (ii) We prove that every "almost integer" sequence $\hat{I}›$, i.e. satisfying $\hat{I}{»}_n=n+\hat{I}{\pm }_n$, $0\ne \hat{I}{\pm }_n\to 0$, is $p$-generating; and (iii) We construct $p$-generating sets $\hat{I}›$ such that the successive differences $\hat{I}{»}_{n+1}-\hat{I}{»}_n$ attain only two different positive values. The constructions are, in a sense, sharp: it is well known that $\hat{I}›$ cannot be Hadamard lacunary and cannot be contained in any arithmetic progression.

To appear in the Journal of the London Mathematical Society