Charles Alba, Nathan Roy
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.
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We generalize the Farey-Brocot partition to a twodimensional continued fraction algorithm and generalized Farey-Brocot nets. We give an asymptotic formula for the moments of order β.
We present the classical Stern-Brocot tree and provide a new proof of the fact that every rational number between 0 and 1 appears in the tree. We then generalize theStern-Brocot tree to allow for arbitrary choice of starting terms, and prove that in all cases the tree maintains the property that every rational number between the two starting terms appears exactly once.
Diophantine quadruples are sets of four distinct positive integers such that the product of any two is one less than a square. All known examples belong to an infinite set which can be constructed recursively. Some observations on these regular solutions are presented. In particular we see how factors of the four numbers satisfy relations which generalise the Stern-Brocot Tree. There is also an update on the search for rational Diophantine sextuples with five new examples.
In this paper we discuss a natural generalization of the Stern Brocot tree 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 $c/d$. Our main result is a determination of which rational numbers between the starting terms appear in the tree. We extend this result to arbitrary reduction schemes as well.
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.
The Stern-Brocot tree and Minkowki's question mark function $?(x)$ (or Conway's box function) are related to the continued fraction expansion of numbers from Q with unary encoding of the partial denominators. We first define binary encodings $C_I, C_{II}$ of the natural numbers, adapted to the Gauß-Kuz'min measure for the distribution of partial denominators. We then define the V$_1$ tree as analogue to the Stern-Brocot tree, using the binary encondings $C_I, C_{II}$. We shall see that all numbers with denominator $q$ are present in the first $3.44\log_2(q)$ levels, instead of $1/q$ appearing …
We find a duality between two well-known trees, the Calkin-Wilf tree and the Stern-Brocot tree, derived from cluster algebra theory. The vertex sets of these trees are the set of rational numbers, and they have cluster structures induced by one-punctured torus. In particular, the Calkin-Wilf tree is an example of the structure given by initial-seed mutations.
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。
We study the paths in the Farey graph going from (1,0) to (0,1) , or the finite subtrees of the Stern-Brocot tree . We show they form an operad and isolate the "coronas", which are especially spiked paths. We show that {(x,y), gcd(x,y)=1 , x+y < R } is a corona.
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.
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.
Denominators in the Farey Sequence, the Stern Sequence, and the Frobenius Problem with Relatively Prime Multipliers
该研究针对带系数互素限制的弗罗贝尼乌斯开放问题，证明此类问题仍存在无法表示的最大整数，并探究其与斯特恩序列等的关联及相关性质。
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…
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.
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。
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.
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。
In this paper we study the properties of the \emph{Triangular tree}, a complete tree of rational pairs introduced in \cite{cas}, in analogy with the main properties of the Farey tree (or Stern-Brocot tree). To our knowledge the Triangular tree is the first generalisation of the Farey tree constructed using the mediant operation. In particular we introduce a two-dimensional representation for the pairs in the tree, a coding which describes how to reach a pair by motions on the tree, and its description in terms of $SL(3,\mathbb{Z})$ matrices. The tree and the properties we study are then used t…
There are two well-known ways to enumerate the positive rational numbers in an infinite binary tree: the Farey/Stern-Brocot tree and the Calkin-Wilf tree. In this brief note, we describe these two trees as `transpose shadows' of a tree of matrices (a result due to Backhouse and Ferreira) via a new proof using yet another famous tree of rationals: the topograph of Conway and Fung.
Continued fractions are linked to Stern's diatomic sequence 0,1,1,2,1,3,2,3,1,4,... (given by the recursion relation a_2n=a_n and a_{2n+1} = a_n + a_{n+1}, where a_0=0 and a_1=1), which has long been known. Using a particular multidimensional continued fraction algorithm (the Farey algorithm), we will generalize the diatomic sequence to a collection of numbers that quite naturally should be called the triatomic sequence (or a two-dimensional Pascal with memory sequence). As continued fractions and the diatomic sequence can be thought of as coming from systematic subdivisions of the unit interv…
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_{…
Parity vectors and paradoxical sequences in the accelerated Collatz map
本文研究了加速科拉茨映射中奇偶向量和悖论序列的性质，通过三个定理和一个数值观察，提出了关于奇偶向量密度、悖论序列计数以及有界长度悖论序列密度零的结论。
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