Refining the Two-Dimensional Signed Small Ball Inequality

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