Open Access
Translator Disclaimer
2021 The Littlewood-Offord problem for Markov chains
Shravas Rao
Author Affiliations +
Electron. Commun. Probab. 26: 1-11 (2021). DOI: 10.1214/21-ECP410

Abstract

The celebrated Littlewood-Offord problem asks for an upper bound on the probability that the random variable ε1v1++εnvn lies in the Euclidean unit ball, where ε1,,εn{1,1} are independent Rademacher random variables and v1,,vnRd are fixed vectors of at least unit length. We extend some known results to the case that the εi 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 vi 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.

Funding Statement

This material is based upon work supported by the National Science Foundation Graduate Research Fellowship Program under Grant No. DGE-1342536.

Citation

Download Citation

Shravas Rao. "The Littlewood-Offord problem for Markov chains." Electron. Commun. Probab. 26 1 - 11, 2021. https://doi.org/10.1214/21-ECP410

Information

Received: 18 December 2020; Accepted: 16 June 2021; Published: 2021
First available in Project Euclid: 26 July 2021

Digital Object Identifier: 10.1214/21-ECP410

Subjects:
Primary: 60F10

Keywords: Littlewood-Offord , Markov chain , pseudorandom generator

JOURNAL ARTICLE
11 PAGES


SHARE
Back to Top