Single radius spherical cap discrepancy via gegenbadly approximable numbers

TL;DR

This paper refines discrepancy bounds for points on spheres, showing that for certain radii, specific sets of Gegenbauer-badly approximable numbers guarantee the existence of spherical caps with large point deviations.

Contribution

It introduces a novel connection between Gegenbauer polynomial approximation and spherical cap discrepancy, extending Beck's result to special radii and number sets.

Findings

Existence of spherical caps with large discrepancy for specific radii

Introduction of Gegenbauer-badly approximable numbers

Refinement of discrepancy bounds for certain dimensions

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.