Almost prime Pythagorean triples in thin orbits

TL;DR

This paper investigates the distribution of Pythagorean triples within certain thin orbits, demonstrating the existence of infinitely many triples with hypotenuse, area, and side products having few prime factors, and providing asymptotic counts.

Contribution

It introduces a new approach to analyze Pythagorean triples in thin orbits, quantifying the prime factorization properties and asymptotic counts of such triples.

Findings

Infinitely many triples with few prime factors in key quantities

Explicit bounds on the number of such triples up to norm T

Asymptotic formula for the count of triples with bounded norm

Abstract

For the ternary quadratic form Q(x) = x^2 + y^2 - z^2 and a non-zero Pythagorean triple x_0 in Z^3 lying on the cone Q(x) = 0, we consider an orbit O = x_0 Gamma of a finitely generated subgroup Gamma < SO_Q(Z) with critical exponent exceeding 1/2. We find infinitely many Pythagorean triples in O whose hypotenuse, area, and product of side lengths have few prime factors, where "few" is explicitly quantified. We also compute the asymptotic of the number of such Pythagorean triples of norm at most T, up to bounded constants.