Pythagorean Triples in the Fibonacci Model Set

Almost prime Pythagorean triples in thin orbits

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.

两色划分正则性毕达哥拉斯三元组的一个新证明

A new proof of two-color partition regularity of Pythagorean triples

本文给出一个由ChatGPT Astra发现的新证明，说明自然数两色染色时必存在单色毕达哥拉斯三元组。

沿广义勾股数的矫饰系统的递归性

Recurrence for pretentious systems along generalized Pythagorean triples

针对沿广义勾股数（方程$ax^2 + b y^2 = c z^2$的解）的矫饰保测乘法作用建立多重递归结果，证实广义勾股分划正则性猜想的遍历理论形式，推论得相关有限着色自然数集含单色广义勾股数。

The dynamics of Pythagorean triples

We construct a piecewise onto 3-to-1 dynamical system on the positive quadrant of the unit circle, such that for rational points (which correspond to normalized Primitive Pythagorean Triples), the associated ternary expansion is finite, and is equal to the address of the PPT on Barning's ternary tree of PPTs, while irrational points have infinite expansions. The dynamical system is conjugate to a modified Euclidean algorithm. The invariant measure is identified, and the system is shown to be conservative and ergodic. We also show, based on a result of Aaronson and Denker, that the dynamical sy…

Billiards on pythagorean triples and their Minkowski functions

It has long been known that the set of primitive pythagorean triples can be enumerated by descending certain ternary trees. We unify these treatments by considering hyperbolic billiard tables in the Poincare disk model. Our tables have m>=3 ideal vertices, and are subject to the restriction that reflections in the table walls are induced by matrices in the triangle group PSU^\pm_{1,1}\Zbb[i]. The resulting billiard map \tilde B acts on the de Sitter space x_1^2+x_2^2-x_3^2=1, and has a natural factor B on the unit circle, the pythagorean triples appearing as the B-preimages of fixed points. We…