Almost Prime Coordinates for Anisotropic and Thin Pythagorean Orbits

TL;DR

This paper enhances the understanding of distribution in Affine Sieve problems, leading to improved bounds on saturation numbers by doubling the exponent of distribution in specific Pythagorean orbit contexts.

Contribution

It introduces a method that doubles the exponent of distribution in certain Affine Sieve problems, improving bounds on saturation numbers.

Findings

Increased the exponent of distribution in specific problems.

Reduced bounds on saturation numbers.

Applicable to Pythagorean orbit problems.

Abstract

We make an observation which doubles the exponent of distribution in certain Affine Sieve problems, such as those considered by Liu-Sarnak, Kontorovich, and Kontorovich-Oh. As a consequence, we decrease the known bounds on the saturation numbers in these problems.