ON THE INCREASING PARTIAL QUOTIENTS OF CONTINUED FRACTIONS OF POINTS IN THE PLANE

For any x in $[0,1)$ , let $[a_1(x),a_2(x),a_3(x),\ldots ]$ be its continued fraction. Let $\psi :\mathbb {N}\to \mathbb {R}^+$ be such that $\psi (n) \to \infty $ as $n\to \infty $ . For any positive integers s and t, we study the set $$ \begin{align*}E(\psi)=\{(x,y)\in [0,1)^2: \max\{a_{sn}(x), a_{tn}(y)\}\ge \psi(n) \ {\text{for all sufficiently large}}\ n\in \mathbb{N}\} \end{align*} $$ and determine its Hausdorff dimension.

A DIMENSIONAL RESULT ON THE PRODUCT OF CONSECUTIVE PARTIAL QUOTIENTS IN CONTINUED FRACTIONS

This paper is concerned with the growth rate of the product of consecutive partial quotients relative to the denominator of the convergent for the continued fraction expansion of an irrational number. More precisely, given a natural number $m,$ we determine the Hausdorff dimension of the following set: $$ \begin{align*} E_m(\tau)=\bigg\{x\in [0,1): \limsup\limits_{n\rightarrow\infty}\frac{\log (a_n(x)a_{n+1}(x)\cdots a_{n+m}(x))}{\log q_n(x)}=\tau\bigg\}, \end{align*} $$ where $\tau $ is a nonnegative number. This extends the dimensional result of Dirichlet nonimprovable sets (when $m=1$ ) shown by Hussain, Kleinbock, Wadleigh and Wang.

ON THE LIMITS OF SOME p-ADIC SCHNEIDER CONTINUED FRACTIONS

In the present paper, we first generalize some convergence results for continued fractions given in real domain and p-adic domain. However, we prove the transcendence of a p-adic number given by it's Schneider continued fractions, such that the sequence of partial quotients is a Thue-Morse sequence.

Farey Sequences for Thin Groups

The Farey sequence is the set of rational numbers with bounded denominator. We introduce the concept of a generalized Farey sequence. While these sequences arise naturally in the study of discrete and thin subgroups, they can be used to study interesting number theoretic sequences—for example rationals whose continued fraction partial quotients are subject to congruence conditions. We show that these sequences equidistribute and the gap distribution converges and answer an associated problem in Diophantine approximation. Moreover, for one example, we derive an explicit formula for the gap distribution. For this example, we construct the analogue of the Gauss measure, which is ergodic for the Gauss map. This allows us to prove a theorem about the associated Gauss–Kuzmin statistics.

A Transcendental Unbounded Continued Fraction Expansions over A Finite Field

Let Fq be a finite field and Fq((X−1 )) the field of formal power series with coefficients in Fq. The purpose of this paper is to exhibit a family of transcendental continued fractions of formal power series over a finite field through some specific irregularities of its partial quotients

A curiosity About (−1)[e] +(−1)[2e] + ··· +(−1)[Ne]

Let α be an irrational real number; the behaviour of the sum SN (α):= (−1)[α] +(−1)[2α] + ··· +(−1)[Nα] depends on the continued fraction expansion of α/2. Since the continued fraction expansion of \sqrt 2 /2 has bounded partial quotients, {S_N}\left( {\sqrt 2 } \right) = O\left( {\log \left( N \right)} \right) and this bound is best possible. The partial quotients of the continued fraction expansion of e grow slowly and thus {S_N}\left( {2e} \right) = O\left( {{{\log {{\left( N \right)}^2}} \over {\log \,\log {{\left( N \right)}^2}}}} \right), again best possible. The partial quotients of the continued fraction expansion of e/2 behave similarly as those of e. Surprisingly enough 1188.

Besicovitch cascades and bounded partial quotients

The article continues a series of works studying cylindrical transformations having discrete orbits (Besicovitch cascades). For any γ ∈ (0,1) and any ɛ > 0 we construct a Besicovitch cascade over some rotation with bounded partial quotients, and with a γ–Hölder function, such that the Hausdorff dimension of the set of points in the circle having discrete orbits is greater than 1 − γ− ɛ.