Sharpness of the s-dimensional spherical cap discrepancy bound

Establish the sharpness of inequality D_{L2,s}^C(X) ≥ c_{d,s} (H_s(X)/a(X,s))^{−(d+1)/(2(d−s))} for lower Ahlfors–David regular sets X ⊂ S^d with H_s(X) ≥ 1, by proving that for every dimension d ≥ 2 and every s with 0 < s < d, the exponent (d+1)/(2(d−s)) in this bound cannot be improved; equivalently, construct families of lower Ahlfors–David regular sets of Hausdorff dimension s whose s-dimensional spherical cap L2-discrepancy attains this order.

Background

The paper extends spherical cap L2-discrepancy from point sets to sets of arbitrary Hausdorff dimension s, defining D_{L2,s}^C(X) in terms of the normalized s-dimensional Hausdorff measure on X. Theorem 4 proves a general lower bound in terms of H_s(X) and the lower Ahlfors–David regularity constant a(X,s): D_{L2,s}^C(X) ≥ c_{d,s} (H_s(X)/a(X,s))^{−(d+1)/(2(d−s))}. This recovers Beck’s bound when s=0 and vanishes as s→d.

The authors note that while the endpoint behavior is consistent with known cases, they have not proven sharpness of the bound for intermediate dimensions, and they explicitly conjecture that the obtained order is sharp for 0 < s < d.