Continued Fractions with Partial Quotients Bounded in Average

In this paper we present a convergence theorem for continued fractions of the form $K_{n=1}^{\infty}a_{n}/1$. By deriving conditions on the $a_{n}$ which ensure that the odd and even parts of $K_{n=1}^{\infty}a_{n}/1$ converge, these same conditions also ensure that they converge to the same limit. Examples will be given.

For $b\in\mathbb{N}, b\ge2$ we determine the limit points of certain subsets of $$ \left\{\frac{b^n\pmod{n}}{n}:n\in\mathbb{N}\right\}. $$ As a consequence, we obtain the density of the latter set in $[0,1]$, a result first established in 2013 by Cilleruelo, Kumchev, Luca, Ru\'{e} and Shparlinski..

Every positive rational number has representations as Egyptian fractions (sums of reciprocals of distinct positive integers) with arbitrarily many terms and with arbitrarily large denominators. However, such representations normally use a very sparse subset of the positive integers up to the largest demoninator. We show that for every positive rational there exist Egyptian fractions whose largest denominator is at most N and whose denominators form a positive proportion of th...

We give bounds on the number of solutions to the Diophantine equation (X+1/x)(Y+1/y) = n as n tends to infinity. These bounds are related to the number of solutions to congruences of the form ax+by = 1 modulo xy.

In 1974, M. B. Nathanson proved that every irrational number $\alpha$ represented by a simple continued fraction with infinitely many elements greater than or equal to $k$ is approximable by an infinite number of rational numbers $p/q$ satisfying $|\alpha-p/q|<1/(\sqrt{k^2+4}q^2)$. In this paper we refine this result.

Several conjectural continued fractions found with the help of various algorithms are published in this paper.

We obtain a new characterization for irrational numbers of constant type -- defined as irrationals with bounded partial quotients in their continued fraction expansion. The result is essential in the formulation of stability criteria for orbits of quantum twist maps in a class of dynamical systems.

By a classical result of Gauss and Kuzmin, the continued fraction expansion of a ``random'' real number contains each digit $a\in\mathbb{N}$ with asymptotic frequency $\log_2(1+1/(a(a+2)))$. We generalize this result in two directions: First, for certain sets $A\subset\mathbb{N}$, we establish simple explicit formulas for the frequency with which the continued fraction expansion of a random real number contains a digit from the set $A$. For example, we show that digits of t...

Let $\omega=[a_1, a_2, \cdots]$ be the infinite expansion of continued fraction for an irrational number $\omega \in (0,1)$; let $R_n (\omega)$ (resp. $R_{n, \, k} (\omega)$, $R_{n, \, k+} (\omega)$) be the number of distinct partial quotients each of which appears at least once (resp. exactly $k$ times, at least $k$ times) in the sequence $a_1, \cdots, a_n$. In this paper it is proved that for Lebesgue almost all $\omega \in (0,1)$ and all $k \geq 1$, $$ \displaystyle \lim_{...

Let $\displaystyle \sum_{i=a}^b \frac{1}{i} = \frac{u_{a,b}}{v_{a,b}}$ with $u_{a,b}$ and $v_{a,b}$ coprime. Erd\H{o}s and Graham asked the following: Does there, for every fixed $a$, exist a $b$ such that $v_{a,b} < v_{a,b-1}$? If so, what is the least such $b = b(a)$? In this paper we will investigate these problems in a more general setting, answer the first question in the affirmative and obtain the bounds $a + 0.54\log(a) < b(a) \le 4.374(a-1)$, which hold for all large ...