Almost prime Pythagorean triples in thin orbits (Q2898924)

For the ternary quadratic form \(Q(x)=x^2+y^2-z^2\) and a non-zero Pythagorean triple \(x_0\in \mathbb{Z}^3\) lying on the cone \(Q(x)=0\), consider an orbit \(\text{O}=x_0\Gamma\) of a finitely generated subgroup \(\Gamma< \mathrm{SO}_Q(\mathbb{Z})\) with critical exponent exceeding \(1/2\). In this paper, infinitely many Pythagorean triples in \(\text{O}\) whose hypotenuse, area and product of side lengths have few prime factors, where ``few'' is explicitly quantified, have been obtained. The asymptotic of the number of such Pythagorean triples of norm at most \(T\), up to bounded constants have also been computed.