Unconditional structures of translates for $L_p(R^d)$

We prove that a sequence $(f_i)_{i=1}^\infty$ of translates of a fixed $f\in L_p(R)$ cannot be an unconditional basis of $L_p(R)$ for any $1\le p<\infty$. In contrast to this, for every $2<p<\infty$, $d\in N$ and unbounded sequence $(\lambda_n)_{n\in N}\subset R^d$ we establish the existence of a function $f\in L_p(R^d)$ and sequence $(g^*_n)_{n\in N}\subset L_p^*(R^d)$ such that $(T_{\lambda_n} f, g^*_n)_{n\in N}$ forms an unconditional Schauder frame for $L_p(R^d)$. In particular, there exists a Schauder frame of integer translates for $L_p(R)$ if (and only if) $2<p<\infty$.