# Generating an equidistributed net on a unit n-sphere using random rotations

## TL;DR

This paper introduces a randomized algorithm that efficiently generates an $ ext{epsilon}$-net on an n-sphere, enabling accurate approximation of integrals of Lipschitz functions with high probability.

## Contribution

It presents a novel randomized method for constructing an $ ext{epsilon}$-net on the sphere using a specific number of random rotations and word combinations, ensuring equidistribution.

## Key findings

- Algorithm succeeds with high probability
- Produces an $ ext{epsilon}$-net with controlled size
- Effective for approximating integrals of Lipschitz functions

## Abstract

We develop a randomized algorithm (that succeeds with high probability) for generating an $\epsilon$-net in a sphere of dimension n. The basic scheme is to pick $O(n \ln(1/n) + \ln(1/\delta))$ random rotations and take all possible words of length $O(n \ln(1/\epsilon))$ in the same alphabet and act them on a fixed point. We show this set of points is equidistributed at a scale of $\epsilon$. Our main application is to approximate integration of Lipschitz functions over an n-sphere.