On the signed small ball inequality

Abstract

Let $h_R$ denote an $L^{\mathrm{\infty }}$-normalized Haar function adapted to a dyadic rectangle $R\subset [0,1{]}^d$. We show that for all choices of coefficients $\alpha (R)\in \{\pm 1\}$, we have the following lower bound on the $L^{\mathrm{\infty }}$-norms of the sums of such functions, where the sum is over rectangles of a fixed volume:
$$n^{\eta (d)}\lesssim \parallel \sum _{|R|=2^{-n}}\alpha (R)h_R(x){\parallel }_{L^{\mathrm{\infty }}([0,1{]}^d)},\text{for all }\eta (d)<\frac{d-1}{2}+\frac{1}{8d},$$
where the implied constant is independent of $n\ge 1$. The inequality above (without restriction on the coefficients) arises in connection to several areas, such as Probabilities, Approximation, and Discrepancy. With $\eta (d)=(d-1)/2$, the inequality above follows from orthogonality, while it is conjectured that the inequality holds with $\eta (d)=d/2$. This is known and proved in $(Talagrand,1994)$ in the case of $d=2$, and recent papers of the authors $(Bilyk\text{ and }Lacey,2006)$, $(Bilyk\text{ et al., 2007})$ prove that in higher dimensions one can take $\eta (d)>(d-1)/2$, without specifying a particular value of $\eta$. The restriction ${\alpha }_R\in \{\pm 1\}$ allows us to significantly simplify our prior arguments and to find an explicit value of $\eta (d)$.