Single radius spherical cap discrepancy via gegenbadly approximable numbers

Abstract

A celebrated result of Beck shows that for any set of $N$ points on ${\mathbb{S}}^d$ there always exists a spherical cap $B\subset {\mathbb{S}}^d$ such that number of points in the cap deviates from the expected value $\sigma (B)\cdot N$ by at least $N^{1/2-1/2d}$, where $\sigma$ is the normalized surface measure. We refine the result and show that, when $d \not\equiv 1 ~(\mbox{mod}~4)$, there exists a (small and very specific) set of real numbers such that for every $r>0$ from the set one is always guaranteed to find a spherical cap $C_r$ with the given radius $r$ for which the result holds. The main new ingredient is a generalization of the notion of badly approximable numbers to the setting of Gegenbauer polynomials: these are fixed numbers $x\in (-1,1)$ such that the sequence of Gegenbauer polynomials $(C_n^{\lambda }(x){)}_{n=1}^{\infty }$ avoids being close to 0 in a precise quantitative sense.