Sums of two squares in short intervals

TL;DR

This paper demonstrates that short intervals and arithmetic progressions contain significantly more numbers expressible as the sum of two squares than average, especially when the interval length is very small.

Contribution

It establishes new lower bounds for the density of sums of two squares in very short intervals and progressions, advancing understanding of their distribution.

Findings

Short intervals contain at least a constant times y^{1/10} sums of two squares.

Results hold for intervals with length y=o((log x)^{5/9}).

Similar results are obtained for short arithmetic progressions.

Abstract

We show that there are short intervals $[x,x+y]$ containing $\gg y^{1/10}$ numbers expressible as the sum of two squares, which is many more than the average when $y=o( (\log{x})^{5/9})$. We obtain similar results for sums of two squares in short arithmetic progressions.