Stern-Brocot Trees from Weighted Mediants

A dichotomy between uniform distributions of the Stern-Brocot and the Farey sequence

We employ infinite ergodic theory to show that the even Stern-Brocot sequence and the Farey sequence are uniformly distributed mod 1 with respect to certain canonical weightings. As a corollary we derive the precise asymptotic for the Lebesgue measure of continued fraction sum-level sets as well as connections to asymptotic behaviours of geometrically and arithmetically restricted Poincaré series. Moreover, we give relations of our main results to elementary observations for the Stern-Brocot tree.

Stern-Brocot 序列与 Minkowski 问号函数的求和

Sums with Stern-Brocot sequences and Minkowski question mark function

本文肯定回答了 Moshchevitin 关于 Stern-Brocot 序列余项在 n 趋于无穷时是否趋于 0 的问题，并给出了关于 Minkowski 问号函数逆函数 Fourier-Stieltjes 系数定理的正确证明与扩展结果。

A multifractal analysis for Stern-Brocot intervals, continued fractions and Diophantine growth rates

In this paper we obtain multifractal generalizations of classical results by Lévy and Khintchin in metrical Diophantine approximations and measure theory of continued fractions. We give a complete multifractal analysis for Stern--Brocot intervals, for continued fractions and for certain Diophantine growth rates. In particular, we give detailed discussions of two multifractal spectra closely related to the Farey map and the Gauss map.

Lacunary formal power series and the Stern-Brocot sequence

Let $F(X) = \sum_{n \geq 0} (-1)^{\varepsilon_n} X^{-λ_n}$ be a real lacunary formal power series, where $\varepsilon_n = 0, 1$ and $λ_{n+1}/λ_n > 2$. It is known that the denominators $Q_n(X)$ of the convergents of its continued fraction expansion are polynomials with coefficients $0, \pm 1$, and that the number of nonzero terms in $Q_n(X)$ is the $n$th term of the Stern-Brocot sequence. We show that replacing the index $n$ by any 2-adic integer $ω$ makes sense. We prove that $Q_ω(X)$ is a polynomial if and only if $ω\in {\mathbb Z}$. In all the other cases $Q_ω(X)$ is an infinite formal powe…

Enumerating the rationals from left to right

Farey sequences, Stern-Brocot sequences, the Calkin-Wilf sequences are shown to be generated via almost identical second order recurrence relations. These sequences have combinatorial, computational, and geometric applications, and are useful for enumerating the rational numbers.

二叉树与两平方和

Binary Trees and Sum of Two Squares

本文引入一个基于矩阵的二叉树，作为Stern-Brocot树和Calkin-Wilf树的基础，并探讨其与连分数的联系，进而给出Brillhart关于两平方和定理的路径表示证明。

Dichotomic random number generators

We introduce several classes of pseudorandom sequences which represent a natural extension of classical methods in random number generation. The sequences are obtained from constructions on labeled binary trees, generalizing the well-known Stern-Brocot tree.

论 Euclid 算法与初等数论

On Euclid's Algorithm and Elementary Number Theory

本文以 Euclid 算法和不变性为统一接口，验证并构造数论结论，提出 gcd 分配性质及两种等价的高效正有理数枚举算法。

On Using (Z^2, +) Homomorphisms to Generate Pairs of Coprime Integers

We use the group $(\Z^2,+)$ and two associated homomorphisms, $τ_0, τ_1$, to generate all distinct, non-zero pairs of coprime, positive integers which we describe within the context of a binary tree which we denote $T$. While this idea is related to the Stern-Brocot tree and the map of relatively prime pairs, the parents of an integer pair these trees do not necessarily correspond to the parents of the same integer pair in $T$. Our main result is a proof that for $x_i \in \{0,1\}$, the sum of the pair $τ_{x_1}τ_{x_2}... τ_{x_n} [1,2]$ is equal to the sum of the pair $τ_{x_n}τ_{x_{n-1}} ... τ_{…