Refining the Two-Dimensional Signed Small Ball Inequality

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, $$\left| \left\{ x \in [0,1)^2: \sum_{|R| = 2^{-n}}{ \varepsilon_R h_R} = n + 1 - 2k\right\} \right| = \frac{1}{2^{n+1}}\binom{n+1}{k}.$$