Year: 2025
Type: article
Abstract: Consider a quadratic polynomial upper Q left parenthesis xi 1 comma ellipsis comma xi Subscript n Baseline right parenthesis Q ( ξ 1 , … , ξ n ) $Q(\xi_{1},\ldots,\xi_{n})$ of independent Rademacher random variables xi 1 comma ellipsis comma xi Subscript n Baseline ξ 1 , … , ξ n $\xi_{1},\ldots,\xi_{n}$ . To what extent can upper Q left parenthesis xi 1 comma ellipsis comma xi Subscript n Baseline right parenthesis Q ( ξ 1 , … , ξ n ) $Q(\xi_{1},\ldots,\xi_{n})$ concentrate on a single value? This quadratic version of the classical Littlewood–Offord problem was popularised by Costello, Tao and Vu in their study of symmetric random matrices. In this paper, we obtain an essentially optimal bound for this problem, as conjectured by Nguyen and Vu. Specifically, if upper Q left parenthesis xi 1 comma ellipsis comma xi Subscript n Baseline right parenthesis Q ( ξ 1 , … , ξ n ) $Q(\xi_{1},\ldots,\xi_{n})$ ‘robustly depends on at least m of the xi Subscript i ξ i $\xi_{i}$ ’ in the sense that there is no way to pin down the value of upper Q left parenthesis xi 1 comma ellipsis comma xi Subscript n Baseline right parenthesis Q
Source: Compositio Mathematica
Authors: Matthew Kwan, Lisa Sauermann
Language: English
Subfield: Statistics and Probability
Field: Mathematics
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $3,450
OpenAlex ID: W7142985866