Littlewood–Offord problem

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]\displaystyle{ \Big( c \, \log n / \sqrt{n} \Big) \, 2^n }[/math] of the 2ⁿ possible subsums of S fall into the disc.

In 1945 Paul Erdős improved the upper bound for d = 1 to

[math]\displaystyle{ {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.

Suppose S = {v₁, …, v_n}. By subtracting

[math]\displaystyle{ \frac{1}{2} \sum_{i = 1}^n v_i }[/math]

from each possible subsum (that is, by changing the origin and then scaling by a factor of 2), the Littlewood–Offord problem is equivalent to the problem of determining the number of sums of the form

[math]\displaystyle{ \sum_{i = 1}^n \varepsilon_i v_i }[/math]

that fall in the target set A, where [math]\displaystyle{ \varepsilon_i }[/math] takes the value 1 or −1. This makes the problem into a probabilistic one, in which the question is of the distribution of these random vectors, and what can be said knowing nothing more about the v_i.