# Solution of the Littlewood-Offord problem in high dimensions

### Abstract

Consider the $2^n$ partial sums of arbitrary $n$ vectors of length at least one in $d$-dimensional Euclidean space. It is shown that as $n$ goes to infinity no closed ball of diameter $\Delta$ contains more than $(|\Delta| +1+o(1)) \binom{n}{|n/2|}$ out of these sums and this is best possible. For $\Delta – |\Delta|$ small an exact formula is given.

Peter Frankl

Zoltán Füredi