On Linnik's conjecture: sums of squares and microsquares

TL;DR

This paper proves that almost all natural numbers (excluding some residues) can be expressed as the sum of three squares, with one square being very small, confirming Linnik's conjecture for most numbers and refining results on points on the sphere.

Contribution

It establishes that nearly all numbers not divisible by 4 or congruent to 7 mod 8 are sums of three squares with a small square, confirming Linnik's conjecture for almost all numbers.

Findings

Almost all such numbers are sums of three squares with a small square.

The result confirms Linnik's conjecture for most natural numbers.

It improves bounds on nearest neighbor distances on the sphere.

Abstract

We show that almost all natural numbers n not divisible by 4, and not congruent to 7 modulo 8, are represented as the sum of three squares, one of which is the square of an integer no larger than (log n)^{1+e} (any e>0). This answers a conjecture of Linnik for almost all natural numbers, and sharpens a conclusion of Bourgain, Rudnick and Sarnak concerning nearest neighbour distances between normalised integral points on the sphere.