On a lemma of Littlewood and Offord

Abstract

In their work on the distribution of roots of algebraic equations, Littlewood and Offord proved in 1943 the following result:

Let a_i, a₂,…, a_n be complex numbers with |a_i| ≥ 1 for all i, and consider the 2ⁿ linear combinations $\sum {_{i = 1}^n{\varepsilon _i}{a_i}}$ with ε_i ∈ {1, -1}. Then the number of sums $\sum {_{i = 1}^n{\varepsilon _i}{a_i}}$ which He in the interior of any circle of radius 1 is not greater than

$$ c\frac{{{2^n}}}{{\sqrt n }}\log n $$

for some constant c > 0.