The Littlewood-Offord Problem for Markov Chains
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The celebrated Littlewood-Offord problem asks for an upper bound on the probability that the random variable ϵ_1 v_1 + ... + ϵ_n v_n lies in the Euclidean unit ball, where ϵ_1, ..., ϵ_n ∈{-1, 1} are independent Rademacher random variables and v_1, ..., v_n ∈R^d are fixed vectors of at least unit length.We extend many known results to the case that the ϵ_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. We also construct a pseudorandom generator for the Littlewood-Offord problem using similar techniques.
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