Spherical Cap $L_2$ Discrepancy -- Blessing of Dimensionality and a Distance-Kernel Variant

arXiv:2604.21340v2 Announce Type: replace Abstract: We prove a dimension-explicit upper bound for the information-complexity (also called the inverse of the discrepancy) of the classical spherical cap $L_2$ discrepancy on the $d$-dimensional sphere $\mathbb{S}^d$. For every fixed $\varepsilon\in(0,1)$, this bound implies that a single point suffices in all sufficiently large dimensions, which shows a ``blessing of dimensionality'' for the associated numerical integration problem. We then introduce a one-parameter family of distance reproducing kernels for the Sobolev space $H^{(d+1)/2}(\mathbb{S}^d)$. For the balanced kernel $K(\boldsymbol{x},\boldsymbol{y})=1-\tfrac{1}{\sqrt{2}}\|\boldsymbol{x}-\boldsymbol{y}\|$, a lower bound together with a random-point upper bound shows that the normalized information-complexity is $\Theta_\varepsilon(d)$ for every fixed $\varepsilon\in(0,1)$. Thus its dependence on $d$ is determined up to constants depending only on $\varepsilon$. Finally, we study the one-sided spherical cap discrepancy obtained by restricting the height parameter to $[0,1]$. We derive its kernel and Stolarsky's invariance principle explicitly. A sharp mean-distance inequality, attained by regular simplices, determines its normalized information-complexity exactly in all sufficiently large dimensions and proves strong polynomial tractability with optimal $\varepsilon$-exponent two.