Completeness of uniformly discrete translates in $L^p(\mathbb{R})$

We construct a real sequence $\{{\lambda }_n{\}}_{n=1}^{\mathrm{\infty }}$ satisfying ${\lambda }_n=n+o(1)$, and a Schwartz function $f$ on $\mathbb{R}$, such that for any $N$ the system of translates $\{f(x-{\lambda }_n)\}$, $n>N$, is complete in the space $L^p(\mathbb{R})$ for every $p>1$. The same system is also complete in a wider class of Banach function spaces on $\mathbb{R}$.