In mathematical field of combinatorial geometry, the Littlewood–Offord problem is the problem of determining the number of subsums of a set of vectors that fall in a given convex set. More formally, if V is a vector space of dimension d, the problem is to determine, given a finite subset of vectors S and a convex subset A, the number of subsets of S whose summation is in A.
The first upper bound for this problem was proven (for d = 1 and d = 2) in 1938 by John Edensor Littlewood and A. Cyril Offord. This Littlewood–Offord lemma states that if S is a set of n real or complex numbers of absolute value at least one and A is any disc of radius one, then not more than <math> \Big( c \, \log n / \sqrt{n} \Big) \, 2^n </math> of the 2<sup>n</sup> possible subsums of S fall into the disc.
In 1945 Paul Erdős improved the upper bound for d = 1 to
:<math>{n \choose \lfloor{n/2}\rfloor} \approx 2^n \, \frac{1}{\sqrt{n</math>
using Sperner's theorem. This bound is sharp; equality is attained when all vectors in S are equal. In 1966, Kleitman showed that the same bound held for complex numbers. In 1970, he extended this to the setting when V is a normed space.