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On the lower bounds for the spherical cap discrepancy

Dmitriy Bilyk, Johann S. Brauchart

TL;DR

The paper provides a new, elementary proof of Beck's lower bound for the spherical cap $L_2$-discrepancy on $\mathbb{S}^d$ and leverages Stolarsky's invariance principle to connect discrepancy with the sum of pairwise distances. It derives powerful geometric corollaries (centroid and frame-potential bounds), extends the Beck-type lower bounds to sets of arbitrary Hausdorff dimension with ADR regularity, and refines the constants via uniform, asymptotic, and conjecture-based analyses. It also generalizes the approach to powers of Euclidean distance and to broader kernels, yielding explicit asymptotic constants and connections to lattice-based conjectures (Epstein zeta values) in multiple dimensions. Overall, the work sharpens the understanding of lower bounds for spherical discrepancy, provides practically tight constants in several dimensions, and offers a versatile framework for discrepancy and energy comparisons across dimensions and dimensions beyond 0-dimensional point sets.

Abstract

We start by providing a very simple and elementary new proof of the classical bound due to J. Beck which states that the spherical cap $\mathbb{L}_2$-discrepancy of any $N$ points on the unit sphere $\mathbb S^d$ in $\mathbb{R}^{d+1}$, $d\geq2$, is at least of the order $N^{-\frac12-\frac{1}{2d}}$. The argument used in this proof leads us to many further new results: estimates of the discrepancy in terms of various geometric quantities, an easy proof of {point-independent} upper estimates for the sum of positive powers of Euclidean distances between points on the sphere, lower bounds for the discrepancy of rectifiable curves and sets of arbitrary Hausdorff dimension. Moreover, refinements of the proof also allow us to obtain explicit values of the constants in the lower discrepancy bound on $\mathbb{S}^d$. The value of the obtained asymptotic constant falls within $3\%$ of the conjectured optimal constant on $\mathbb S^2$ (and within up to $7\%$ on $\mathbb S^4$, $\mathbb S^8$, $\mathbb S^{24}$).

On the lower bounds for the spherical cap discrepancy

TL;DR

The paper provides a new, elementary proof of Beck's lower bound for the spherical cap -discrepancy on and leverages Stolarsky's invariance principle to connect discrepancy with the sum of pairwise distances. It derives powerful geometric corollaries (centroid and frame-potential bounds), extends the Beck-type lower bounds to sets of arbitrary Hausdorff dimension with ADR regularity, and refines the constants via uniform, asymptotic, and conjecture-based analyses. It also generalizes the approach to powers of Euclidean distance and to broader kernels, yielding explicit asymptotic constants and connections to lattice-based conjectures (Epstein zeta values) in multiple dimensions. Overall, the work sharpens the understanding of lower bounds for spherical discrepancy, provides practically tight constants in several dimensions, and offers a versatile framework for discrepancy and energy comparisons across dimensions and dimensions beyond 0-dimensional point sets.

Abstract

We start by providing a very simple and elementary new proof of the classical bound due to J. Beck which states that the spherical cap -discrepancy of any points on the unit sphere in , , is at least of the order . The argument used in this proof leads us to many further new results: estimates of the discrepancy in terms of various geometric quantities, an easy proof of {point-independent} upper estimates for the sum of positive powers of Euclidean distances between points on the sphere, lower bounds for the discrepancy of rectifiable curves and sets of arbitrary Hausdorff dimension. Moreover, refinements of the proof also allow us to obtain explicit values of the constants in the lower discrepancy bound on . The value of the obtained asymptotic constant falls within of the conjectured optimal constant on (and within up to on , , ).

Paper Structure

This paper contains 15 sections, 8 theorems, 105 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. A simple proof of Beck's lower bound
  3. Geometric corollaries
  4. Discrepancy of curves and sets of arbitrary Hausdorff dimension.
  5. Refinements: Analysis of the constant $c_d$ in \ref{['eq:Beck']}
  6. Uniform bound
  7. Asymptotic constant
  8. Comparison with conjectured asymptotic behaviour
  9. The case $d=2$
  10. The case $d=4$
  11. The case $d=8$ and $d = 24$
  12. Further remarks
  13. Powers of the Euclidean distance and Generalizations
  14. Proof of Lemmas \ref{['lem:1st.non.negativity']} and \ref{['lem:2nd.inner.product.integrals.asymptotics']}
  15. Proof of Lemma \ref{['lem:sum.asymptotics.alpha']}

Key Result

Lemma 1

Let $d \geq 2$. For all $\mathbf{x}_1, \dots, \mathbf{x}_N \in \mathbb S^d$ and for any $m \ge 0$,

Figures (3)

  • Figure 1: The behaviour of $c_d^{***}$ as a function of (continuous) $d$. Points indicate the values of $c_d^{\mathrm{conj}}$ for $d = 2, 4, 8, 16$.
  • Figure 2: The behaviour of $N^{\frac{3}{2}}\frac{-{\left(-\frac{1}{2}\right)_{m}}}{m!} S(m)$ versus $\frac{1}{N} \leq \frac{m}{N} \leq \log N$ for spherical Fibonacci points on $\mathbb{S}^2$ with $N = 377$ (i.e., $N = F_n$ for $n = 14$) points. There is different behaviour for even and odd $m$.
  • Figure 3: Comparison of the conjectured constant $c_{\alpha,d}^{\mathrm{conj}}$ with the asymptotic constant $c_{\alpha,d}^{\mathrm{asymp}}$ for $0\leq \alpha \leq 2$ for $d = 2$ (top left), $d = 4$ (top right), $d = 8$ (bottom left), and $d = 16$ (bottom right). Inserts provide information about the increasing relative error $\frac{c_{\alpha,d}^{\mathrm{conj}}-c_{\alpha,d}^{\mathrm{asymp}}}{c_{\alpha,d}^{\mathrm{conj}}}$ and the error $c_{\alpha,d}^{\mathrm{conj}}-c_{\alpha,d}^{\mathrm{asymp}}$.

Theorems & Definitions (13)

  • Lemma 1
  • Lemma 2
  • Proposition 3
  • Remark
  • Proposition 4
  • Theorem 5
  • proof
  • Corollary 6
  • Lemma 7
  • Proposition 8
  • ...and 3 more