On the Signed Small Ball Inequality

where the implied constant is independent of n ≥ 1. The inequality above (without restriction on the coefficients) arises in connection to several areas, such as Probabilities, Approximation, and Discrepancy. With η(d) = (d − 1)/2, the inequality above follows from orthogonality, while it is conjectured that the inequality holds with η(d) = d/2. This is known and proved in (Talagrand, 1994) in the case of d = 2, and recent papers of the of the authors (Bilyk and Lacey, 2006), (Bilyk et al., 2007) prove that in higher dimensions one can take η(d) > (d − 1)/2, without specifying a particular value of η. The restriction αR ∈ {±1} allows us to significantly simplify our prior arguments and to find an explicit value of η(d).