Refining the Two-Dimensional Signed Small Ball Inequality

TL;DR

This paper refines the understanding of the two-dimensional signed small ball inequality by precisely characterizing the distribution of the sum of Haar functions over dyadic rectangles with all sign choices.

Contribution

It provides an exact formula for the measure of the set where the sum of Haar functions equals specific values, enhancing the theoretical understanding of the inequality.

Findings

Exact measure of level sets for the sum of Haar functions.

Distribution of the sum follows a binomial pattern.

Improves previous bounds by precise characterization.

Abstract

The two-dimensional signed small ball inequality states that for all possible choices of signs, $$ \left\| \sum_{|R| = 2^{-n}}{ \varepsilon_R h_R} \right\|_{L^{\infty}} \gtrsim n,$$ where the summation runs over all dyadic rectangles in the unit square and $h_R$ denotes the associated Haar function. This inequality first appeared in the work of Talagrand, and alternative proofs are due to Temlyakov and Bilyk & Feldheim (who showed that the supremum equals $n+1$ in all cases). We prove that for all integers $0\leq k \leq n+1$ and all possible choices of signs, $$ \left| \left\{ x \in [0,1)^2: \sum_{|R| = 2^{-n}}{ \varepsilon_R h_R} = n + 1 - 2k\right\} \right| = \frac{1}{2^{n+1}}\binom{n+1}{k}.$$