On the Signed Small-Ball Inequality with Restricted Coefficients

Let R = R-1 x R-2 x ... x R-d denote a dyadic rectangle in the unit cube [0,1](d), d >= 3. Let hR be the normalized Haar function supported on R. We show that for all integers n >= 1, the conjectured signed small-ball inequality parallel to Sigma(vertical bar R vertical bar=2-n) alpha(R)h(R)parallel to(infinity) greater than or similar to n(d/2) where alpha(R) is an element of {+/- 1} holds under the additional assumption that the coefficients alpha(R) also satisfy the "splitting property," namely, alpha(R) = alpha(R1) . alpha(R2xR3x ... xRd), with alpha R-1, alpha(R2xR3x ... xRd) is an element of {+/- 1}. The unrestricted small-ball inequality has connections to multiple fields, such as probability, approximation, and discrepancy.