Optimal dimensional exponent for singleton-partition anti-concentration

Determine whether the dimensional exponent d−1/2 in the all-center, oscillation-normalized anti-concentration estimate for degree-d multi-affine polynomials under the singleton admissible partition can be reduced, thereby identifying the optimal exponent in this setting.

Background

For a degree-d multi-affine polynomial with n active variables, the singleton partition yields the scale n^{d−1/2} in the product-profile small-ball estimate. The block-product examples show that no exponent below d/2 can hold uniformly among all-center, oscillation-normalized estimates with the product profile Φ_d.

The paper does not establish whether the singleton-partition exponent d−1/2 is sharp or can be improved. The authors explicitly identify the optimal exponent for this setting as unknown, making this an unresolved quantitative question about the dimensional dependence of the anti-concentration bound.