The Littlewood-Offord Problem for Markov Chains

The celebrated Littlewood-Offord problem asks for an upper bound on the probability that the random variable $\epsilon_1 v_1 + \cdots + \epsilon_n v_n$ lies in the Euclidean unit ball, where $\epsilon_1, \ldots, \epsilon_n \in \{-1, 1\}$ are independent Rademacher random variables and $v_1, \ldots, v_n \in \mathbb{R}^d$ are fixed vectors of at least unit length.We extend many known results to the case that the $\epsilon_i$ are obtained from a Markov chain, including the general bounds first shown by Erd\H{o}s in the scalar case and Kleitman in the vector case, and also under the restriction that the $v_i$ are distinct integers due to S\'ark\"ozy and Szemeredi. In all extensions, the upper bound includes an extra factor depending on the spectral gap. We also construct a pseudorandom generator for the Littlewood-Offord problem using similar techniques.