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

A real sequence $\mathrm{\Lambda }=\{{\lambda }_n{\}}_{n=1}^{\mathrm{\infty }}$ is called $p$-generating if there exists a function $g$ whose translates $\{g(x-{\lambda }_n){\}}_{n=1}^{\mathrm{\infty }}$ 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 $\mathrm{\Lambda }$ of positive real numbers can be very sparse, namely, the ratios ${\lambda }_{n+1}\mathrm{/}{\lambda }_n$ may tend to $1$ arbitrarily slowly; (ii) We prove that every “almost integer” sequence $\mathrm{\Lambda }$, i.e. satisfying ${\lambda }_n=n+{\alpha }_n$, $0\mathrm{\ne }{\alpha }_n\to 0$, is $p$-generating; and (iii) We construct $p$-generating sets $\mathrm{\Lambda }$ such that the successive differences ${\lambda }_{n+1}-{\lambda }_n$ attain only two different positive values. The constructions are, in a sense, sharp: it is well known that $\mathrm{\Lambda }$ cannot be Hadamard lacunary and cannot be contained in any arithmetic progression.