Almost prime Pythagorean triples in thin orbits

2010/01/03 by Alex Kontorovich, Hee Oh, Kontorovich, Alex +1 · 1 citation
Mathematics · #11E25 #11F72 #11N36 #20H10 #22E40 #Dynamical Systems (math.DS) #FOS: Mathematics #Number Theory (math.NT) #math.DS #math.NT #msc:11E25 #msc:11F72 #msc:11N36 #msc:20H10 #msc:22E40

paper · pdf · doi:10.48550/arxiv.1001.0370

37 pages, 8 figures, 1 table

arxiv created 2010/01/03 · arxiv updated 2010/01/14

Abstract

For the ternary quadratic form Q(x) = x² + y² - z² and a non-zero Pythagorean triple x₀ in Z³ lying on the cone Q(x) = 0, we consider an orbit O = x₀ 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.

Citations

• Uniform expansion bounds for Cayley graphs of SL 2 (𝔽 p )

Cited by

• An Analogue of the Dedekind Eta Function for Hecke Groups H(√(D))

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