Refining the Two-Dimensional Signed Small Ball Inequality

Abstract

The two-dimensional signed small ball inequality states that for all possible choices of signs, $$\begin{aligned} \left\| \sum _{|R| = 2^{-n}}{ \varepsilon _R h_R} \right\| _{L^{\infty }} \gtrsim n, \end{aligned}$$ 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 a stronger result: for all integers $$0\le k \le n+1$$ , all possible choices of signs, and all dyadic rectangles Q with $$|Q| \ge 2^{-n-1}$$ , $$\begin{aligned} \left| \left\{ x \in Q: \sum _{|R| = 2^{-n}}{ \varepsilon _R h_R} = n + 1 - 2k\right\} \right| = \frac{|Q|}{2^{n+1}}\left( {\begin{array}{c}n+1 k\end{array}}\right) . \end{aligned}$$