Almost prime Pythagorean triples in thin orbits

By Alex Kontorovich, H. Oh. 2010

49 citations recorded by the source corpus.

Abstract For the ternary quadratic form Q(x) = x2 + y2 − z2 and a non-zero Pythagorean triple x0 ∈ ℤ3 lying on the cone Q(x) = 0, we consider an orbit 𝒪 = x0Γ of a finitely generated subgroup Γ < SOQ(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.

Subfields: Algebra and Number Theory, Discrete Mathematics and Combinatorics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/d83893e292a1ac7f12f54a90a1b1fba736e620d0

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In Algebra and Number Theory

In Discrete Mathematics and Combinatorics

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Citations in the corpus

Apollonian circle packings and closed horospheres on hyperbolic 3-manifolds (2008, methodology): “In [22], we apply the methods of this paper to the problem of thin orbits of Pythagorean triples having few prime factors.”
Matrix coefficients, counting and primes for orbits of geometrically finite groups (2012, methodology): “19 was previously obtained for n = 2, 3 and s ≤ 0 ([5], [33], [35], [34], [41]).”
ON THE DEVIATION OF ERGODIC AVERAGES FOR HOROCYCLE FLOWS (2013, unknown): “Recently also effective results on the asymptotic equidistribution of closed horocycle orbits for noncofinite (but finitely generated) Γ have been obtained, by Kontorovich and Oh [16] and Lee and Oh [18].”