  We give a new bound on the probability that the random sum $\xi_1 v_1 +...+
\xi_n v_n$ belongs to a ball of fixed radius, where the $\xi_i$ are iid
Bernoulli random variables and the $v_i$ are vectors in $\R^d$. As an
application, we prove a conjecture of Frankl and F\"uredi (raised in 1988),
which can be seen as the high dimensional version of the classical
Littlewood-Offord-Erd\H os theorem.
