The Affine Sieve Beyond Expansion I: Thin Hypotenuses

TL;DR

This paper advances the Affine Sieve method by proving a higher level of distribution for hypotenuses in thin Pythagorean orbits, leading to fewer almost primes than previous results, using bilinear forms and dispersion techniques.

Contribution

It unconditionally establishes a new exponent of distribution beyond current techniques for the Affine Sieve applied to hypotenuses, reducing the almost primes count.

Findings

Proves exponent alpha < 7/24 unconditionally

Produces R = 4 almost primes in the sieve problem

Develops bilinear forms and dispersion methods for incomplete sums

Abstract

We study an instance of the Affine Sieve, producing a level of distribution beyond that which can be obtained from current techniques, even assuming a Selberg/Ramanujan-type spectral gap. In particular, we consider the set of hypotenuses in a thin orbit of Pythagorean triples. Previous work [Kon07, Kon09, KO12] gave an exponent of distribution alpha < 1/12 coming from Gamburd's [Gam02] gap theta = 5/6, thereby producing R = 13 almost primes in this linear sieve problem (see Sec. 1 for definitions). If conditioned on a best possible gap theta = 1/2, the known method would give an exponent alpha < 1/4, and R = 5 almost primes. The exponent 1/4 is the natural analogue of the "Bombieri-Vinogradov" range of distribution for this problem, see Remark 1.19. In this paper, we unconditionally prove the exponent alpha < 7/24 (in the "Elliott-Halberstam" range), thereby producing R = 4 almost primes. The main tools involve developing bilinear forms and the dispersion method in the range of incomplete sums for this Affine Sieve problem.