On the signed small ball inequality

Let \( h_R \) denote an \( L^\infty \)-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 \( L^\infty \)-norms of the sums of such functions, where the sum is over rectangles of a fixed volume:
\[
n^{\eta(d)} \lesssim \Bigg\| \sum_{|R| = 2^{-n}} \alpha(R) h_R(x) \Bigg\|_{L^\infty([0,1]^d)}, \quad \text{for all } \eta(d) < \frac{d-1}{2} + \frac{1}{8d},
\]
where the implied constant is independent of \( n \geq 1 \). 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 \( \eta(d) \).