  We construct a real sequence $\{\lambda_n\}_{n=1}^{\infty}$ 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 $\mathbb{R}$.
