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arXiv · 2308.00694

Single radius spherical cap discrepancy via gegenbadly approximable numbers

Dmitriy Bilyk·Michelle Mastrianni·Stefan Steinerberger
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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.

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Dmitriy Bilyk, Michelle Mastrianni, Stefan Steinerberger. 2023-08-01. Single radius spherical cap discrepancy via gegenbadly approximable numbers. https://arxiv.org/abs/2308.00694

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