On probability measures arising from lattice points on circles

TL;DR

This paper studies probability measures derived from lattice points on circles, characterizing their structure, symmetries, and limitations, revealing a complex, fractal-like geometry influenced by prime powers.

Contribution

It identifies the set of attainable measures, shows their convex and symmetry properties, and uncovers their fractal structure and prime power influence.

Findings

Attainable measures include all extreme symmetric measures.

The set of attainable measures is closed under convolution.

Singularities in the measure set have a fractal structure, absent for square-free radii.

Abstract

A circle, centered at the origin and with radius chosen so that it has non-empty intersection with the integer lattice $\mathbb{Z}^{2}$, gives rise to a probability measure on the unit circle in a natural way. Such measures, and their weak limits, are said to be attainable from lattice points on circles. We investigate the set of attainable measures and show that it contains all extreme points, in the sense of convex geometry, of the set of all probability measures that are invariant under some natural symmetries. Further, the set of attainable measures is closed under convolution, yet there exist symmetric probability measures that are not attainable. To show this, we study the geometry of projections onto a finite number of Fourier coefficients and find that the set of attainable measures has many singularities with a "fractal" structure. This complicated structure in some sense arises from prime powers - singularities do not occur for circles of radius $\sqrt{n}$ if $n$ is square free.