Ford circles, continued fractions, and best approximation of the second kind

The Convergence and Divergence of $q$-Continued Fractions outside the Unit Circle

We consider two classes of $q$-continued fraction whose odd and even parts are limit 1-periodic for $|q|>1$, and give theorems which guarantee the convergence of the continued fraction, or of its odd- and even parts, at points outside the unit circle.

阿波罗尼奥斯圆填充、连分数及其他薄轨道算术的图解导引

An illustrated introduction to the arithmetic of Apollonian circle packings, continued fractions, and other thin orbits

本文整理并扩展两次暑期学校讲义，以阿波罗尼奥斯圆填充、薄群、连分数等为线索，图解数论与几何、动力学及整数轨道之间的联系。

On the Divergence of the Rogers-Ramanujan Continued Fraction on the Unit Circle

Let the continued fraction expansion of any irrational number $t \in (0,1)$ be denoted by $[0,a_{1}(t),a_{2}(t),...]$ and let the i-th convergent of this continued fraction expansion be denoted by $c_{i}(t)/d_{i}(t)$. Let \[ S=\{t \in (0,1): a_{i+1}(t) \geq ϕ^{d_{i}(t)} \text{infinitely often}\}, \] where $ϕ= (\sqrt{5}+1)/2$. Let $Y_{S} =\{\exp(2 πi t): t \in S \}$. It is shown that if $y \in Y_{S}$ then the Rogers-Ramanujan continued fraction, R(y), diverges at y. S is an uncountable set of measure zero. It is also shown that there is an uncountable set of points, $G \subset Y_{S}$, such that…

Asymptotics and Sequential Closures of Continued Fractions and Generalizations

Given a sequence of complex square matrices, $a_n$, consider the sequence of their partial products, defined by $p_n=p_{n-1}a_{n}$. What can be said about the asymptotics as $n\to\infty$ of the sequence $f(p_n)$, where $f$ is a continuous function? A special case of our most general result addresses this question under the assumption that the matrices $a_n$ are an $l_1$ perturbation of a sequence of matrices with bounded partial products. We apply our theory to investigate the asymptotics of the approximants of continued fractions. In particular, when a continued fraction is $l_1$ limit 1-peri…

Convergence properties of the classical and generalized Rogers-Ramanujan continued fraction

The aim of this paper is to study the convergence and divergence of the Rogers-Ramanujan and the generalized Rogers-Ramanujan continued fractions on the unit circle. We provide an example of an uncountable set of measure zero on which the Rogers-Ramanujan continued fraction $R(x)$ diverges and which enlarges a set previously found by Bowman and Mc Laughlin. We further study the generalized Rogers-Ramanujan continued fractions $R_a(x)$ for roots of unity $a$ and give explicit convergence and divergence conditions. As such, we extend some work of Huang towards a question originally investigated …

On Salem numbers, expansive polynomials and Stieltjes continued fractions

A converse method to the Construction of Salem (1945) of convergent families of Salem numbers is investigated in terms of an association between Salem polynomials and Hurwitz quotients via expansive polynomials of small Mahler measure. This association makes use of Bertin-Boyd's Theorem A (1995) of interlacing of conjugates on the unit circle; in this context, a Salem number $β$ is produced and coded by an m-tuple of positive rational numbers characterizing the (SITZ) Stieltjes continued fraction of the corresponding Hurwitz quotient (alternant). The subset of Stieltjes continued fractions ove…

Intrinsic Diophantine approximation on the unit circle and its Lagrange spectrum

Let $\mathscr{L}(S^1)$ be the Lagrange spectrum arising from intrinsic Diophantine approximation on the unit circle $S^1$ by its rational points. We give a complete description of the structure of $\mathscr{L}(S^1)$ below its smallest accumulation point. To this end, we use digit expansions of points on $S^1$, which were originally introduced by Romik in 2008 as an analogue of simple continued fraction of a real number. We prove that the smallest accumulation point of $\mathscr{L}(S^1)$ is 2. Also we characterize the points on $S^1$ whose Lagrange numbers are less than 2 in terms of Romik's di…

An Analytic Heuristic for Multiplicity Computation for Zaremba's Conjecture

Zaremba's Conjecture concerns the formation of continued fractions with partial quotients restricted to a given alphabet. In order to answer the numerous questions that arrive from this conjecture, it is best to consider a semi-group, often denoted $Γ_{A}$, which arises naturally as a subset of $SL_2(\mathbb{Z})$ when considering finite continued fractions. To translate back from this semi-group into rational numbers, we select a projection mapping satisfying certain criteria to recover the numerator and denominator of the continued fractions in rational form. The central question of our resea…

丢番图逼近中的和、级数与乘积

Sums, series, and products in Diophantine approximation

本文围绕丢番图逼近中的和、级数与乘积展开，利用连分数对几乎所有x建立相关结果，阐述其在圆法奇异级数中的应用，并给出Hardy和Littlewood一项经典结论的全新详细证明。