Refining the Two-Dimensional Signed Small Ball Inequality

Abstract

The two-dimensional signed small ball inequality states that for all possible choices of signs, $${\Vert \sum _{|R|=2^{-n}}{\epsilon }_R{h}_R\Vert }_{L^{\infty }}\gtrsim n,$$ 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 $0\le k\le n+1$ and all possible choices of signs, $$\left|\left\{x\in [0,1{)}^2:\sum _{|R|=2^{-n}}{\epsilon }_R{h}_R=n+1-2k\right\}\right|=\frac{1}{2^{n+1}}\left(\genfrac{}{}{0ex}{}{n+1}{k}\right).$$