Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Average variance bounds for integer points on the sphere

Christopher Lutsko

Abstract

Consider the integer points lying on the sphere of fixed radius projected onto the unit sphere. Duke showed that, on congruence conditions for the radius squared, these points equidistribute. To further this study of equidistribution, we consider the variance of the number of points in a spherical cap. An asymptotic for this variance was conjectured by Bourgain-Rudnick-Sarnak. We prove an upper bound of the correct size on the average (over radii) of these variances.

Average variance bounds for integer points on the sphere

Abstract

Consider the integer points lying on the sphere of fixed radius projected onto the unit sphere. Duke showed that, on congruence conditions for the radius squared, these points equidistribute. To further this study of equidistribution, we consider the variance of the number of points in a spherical cap. An asymptotic for this variance was conjectured by Bourgain-Rudnick-Sarnak. We prove an upper bound of the correct size on the average (over radii) of these variances.
Paper Structure (12 sections, 7 theorems, 106 equations)

This paper contains 12 sections, 7 theorems, 106 equations.

Table of Contents

  1. Introduction
  2. Strategy of proof:
  3. Smoothing
  4. Reducing to bounds on Fourier coefficients of automorphic forms
  5. Fourier coefficients of holomorphic cusp forms
  6. Uniform bound
  7. $L^2$ bounds on $\theta_{k,j}$
  8. Bounds on $L$-series
  9. Bounds on the critical line
  10. Bounds on $\mathcal{F}_X$
  11. Proof of Theorem \ref{['thm:smooth']}
  12. Averages over windows

Key Result

Theorem 2

Fix an integer $X$ and a spherical cap $\Omega_X$ with area $\sigma(\Omega_X) = c N_X^\delta$, where $-1< \delta < 0$ and $c>0$ a constant. Then we have Moreover, for any $\frac{X^{3/4}}{\sigma(X)^{3/4}} < H < \infty$ we have

Theorems & Definitions (14)

  • Conjecture 1: BRS
  • Theorem 2
  • Theorem 3
  • proof : Proof of Theorem \ref{['thm:main']}
  • Lemma 4
  • proof
  • Lemma 5
  • proof
  • Lemma 6
  • proof
  • ...and 4 more