Erdős–Littlewood–Offord problem with arbitrary probabilities

Abstract

The classical Erdős–Littlewood–Offord problem concerns the random variable X=a1ξ1+…+anξn, where ai∈R∖{0} are fixed and ξi∼Ber(1/2) are independent. The Erdős–Littlewood–Offord theorem states that the maximum possible concentration probability maxx∈R⁡Pr⁡(X=x) is (n⌊n/2⌋)/2n, achieved, for example, when the ai are all 1. As proposed by Fox, Kwan, and Sauermann, we investigate the general case where ξi∼Ber(p) instead. Using purely combinatorial techniques, we show that the exact maximum concentration probability is achieved for some choice of the coefficients ai with ai∈{−1,1} for all i. Then, using Fourier-analytic techniques, we investigate the optimal ratio of 1s to −1s. Surprisingly, we find that in some cases, the numbers of 1s and −1s can be far from equal.