Erdos-Littlewood-Offord problem with arbitrary probabilities

TL;DR

This paper extends the Erdős-Littlewood-Offord problem to arbitrary Bernoulli probabilities, showing that maximum concentration occurs with ±1 coefficients and analyzing the optimal ratio of these coefficients.

Contribution

It proves that the maximum concentration probability is achieved with coefficients in ext{-1, 1} and explores the optimal ratio of these coefficients for general Bernoulli parameters.

Findings

Maximum concentration achieved with coefficients in ext{-1, 1}

Optimal ratio of 1s to -1s varies and can be far from equal

Purely combinatorial and Fourier-analytic techniques used

Abstract

The classical Erd\H{o}s-Littlewood-Offord problem concerns the random variable $X = a_1 \xi_1 + \dots + a_n \xi_n$, where $a_i \in \mathbb{R} \setminus \{0\}$ are fixed and $\xi_i \sim \text{Ber}(1/2)$ are independent. The Erd\H{o}s-Littlewood-Offord theorem states that the maximum possible concentration probability $\max_{x \in \mathbb{R}} \Pr(X = x)$ is $\binom{n}{\lfloor n/2\rfloor} / 2^n$, achieved when the $a_i$ are all $1$. As proposed by Fox, Kwan, and Sauermann, we investigate the general case where $\xi_i \sim \text{Ber}(p)$ instead. Using purely combinatorial techniques, we show that the exact maximum concentration probability is achieved when $a_i \in \{-1, 1\}$ for each $i$. Then, using Fourier-analytic techniques, we investigate the optimal ratio of $1$s to $-1$s. Surprisingly, we find that in some cases, the numbers of $1$s and $-1$s can be far from equal.