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

A real sequence $\Lambda = \{\lambda_n\}_{n=1}^\infty$ is called $p$-generating if there exists a function $g$ whose translates $\{g(x-\lambda_n)\}_{n=1}^\infty$ 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 $\Lambda$ of positive real numbers can be very sparse, namely, the ratios $\lambda_{n+1} / \lambda_n$ may tend to $1$ arbitrarily slowly; (ii) We prove that every "almost integer" sequence $\Lambda$, i.e. satisfying $\lambda_n = n + \alpha_n$, $0 \neq \alpha_n \to 0$, is $p$-generating; and (iii) We construct $p$-generating sets $\Lambda$ 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 $\Lambda$ cannot be Hadamard lacunary and cannot be contained in any arithmetic progression.