The Littlewood-Offord problem for Markov chains

Abstract

The celebrated Littlewood-Offord problem asks for an upper bound on the probability that the random variable ${\mathit{\epsilon }}_1{v}_1+\cdots +{\mathit{\epsilon }}_n{v}_n$ lies in the Euclidean unit ball, where ${\mathit{\epsilon }}_1,\dots ,{\mathit{\epsilon }}_n\in \{-1,1\}$ are independent Rademacher random variables and $v_1,\dots ,v_n\in {\mathbb{R}}^d$ are fixed vectors of at least unit length. We extend some known results to the case that the ${\mathit{\epsilon }}_i$ 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 Sárközy and Szemeredi. In all extensions, the upper bound includes an extra factor depending on the spectral gap and additional dependency on the dimension. We also construct a pseudorandom generator for the Littlewood-Offord problem using similar techniques.