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 Erdős in the scalar case and Kleitman in the vector case, and also under the restriction that the \(v_i\) are distinct integers due to Sárk\"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.