A non-uniform Littlewood-Offord inequality

Consider a sum $S_n=v_i{\epsilon }_1+\cdots +v_n{\epsilon }_n$, where $(v_i{)}_{i=1}^n$ are non-zero vectors in ${\mathbb{R}}^d$ and $({\epsilon }_i{)}_{i=1}^n$ are independent Rademacher random variables (i.e., $\mathbb{P}({\epsilon }_i=\pm 1)=1\mathrm{/}2$). The classical Littlewood-Offord problem asks for the best possible upper bound for $~{\sup_{x}$\mathbb{P}$(S_n = x)}$. In this paper we consider a non-uniform version of this problem. Namely, we obtain the optimal bound for $\mathbb{P}(S_n=x)$ in terms of the length of the vector $x\in {\mathbb{R}}^d$.