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

The Farey Sequence, Stern Brocot Tree and Euclids Theorem

Farey's sequence is a well-known procedure used to generate proper fractions from 0 to 1. Farey sequence is commonly used in rational approximations of irrational numbers, ford circles and in Riemann hypothesis. Thus, in this paper, we aim to use properties of the Farey's sequence to prove the popular gcd theorem.

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。

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…

Weighted Mediants and Fractals

In this paper we study a natural generalization of the Stern-Brocot sequences which comes from the introduction of weighted mediants. We focus our attention on the case $k = 3$, in which $(2a + c)/(2b + d)$ and $(a + 2c)/(b + 2d)$ are the two mediants inserted between $a/b$ and $c/d$. We state and prove several properties about the cross-differences of Stern-Brocot sequences with $k = 3$, and give a proof of the fractal-like rule that describes the cross-differences of the unit $k = 3$ Stern-Brocot sequences, i.e. the one with usual starting terms $0/1, 1/1$ and with reduction of fractions.

Pell's equation without irrational numbers

We solve Pell's equation in a simple way without continued fractions or irrational numbers, and relate the algorithm to the Stern Brocot tree.

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.

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。

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。