Radical bound for Zaremba's conjecture

We show that, for any fixed $\varepsilon > 0$ and almost all primes $p$, the $g$-ary expansion of any fraction $m/p$ with $\gcd(m,p) = 1$ contains almost all $g$-ary strings of length $k < (5/24 - \varepsilon) \log_g p$. This complements a result of J. Bourgain, S. V. Konyagin, and I. E. Shparlinski that asserts that, for almost all primes, all $g$-ary strings of length $k < (41/504 -\varepsilon) \log_g p$ occur in the $g$-ary expansion of $m/p$.

Even though Zaremba's conjecture remains open, Bourgain and Kontorovich solved the problem for a full density subset. Nevertheless, there are only a handful of explicit sequences known to satisfy the strong version of the conjecture, all of which were obtained using essentially the same algorithm. In this note, we provide a refined algorithm using the folding lemma for continued fractions, which both generalizes and improves on the old one. As a result, we uncover new example...

Legendre's theorem states that every irreducible fraction $\frac{p}{q}$ which satisfies the inequality $\left |\alpha-\frac{p}{q} \right | < \frac{1}{2q^2}$ is convergent to $\alpha$. Later Barbolosi and Jager improved this theorem. In this paper we refine these results.

Given $b=-A\pm i$ with $A$ being a positive integer, we can represent any complex number as a power series in $b$ with coefficients in $\mathcal A=\{0,1,\ldots, A^2\}$. We prove that, for any real $\tau\geq 2$ and any non-empty proper subset $J(b)$ of $\mathcal A$, there are uncountably many complex numbers (including transcendental numbers) that can be expressed as a power series in $b$ with coefficients in $J(b)$ and with the irrationality exponent (in terms of Gaussian int...

For a positive integer $N$ and $\mathbb{A}$ a subset of $\mathbb{Q}$, let $\mathbb{A}$-$\mathcal{KS}(N)$ denote the set of $\alpha=\dfrac{\alpha_{1}}{\alpha_{2}}\in \mathbb{A}\setminus \{0,N\}$ verifying $\alpha_{2}p-\alpha_{1}$ divides $\alpha_{2}N-\alpha_{1}$ for every prime divisor $p$ of $N$. The set $\mathbb{A}$-$\mathcal{KS}(N)$ is called the set of Korselt bases of $N$ in $\mathbb{A}$ or simply the $\mathbb{A}$-Korselt set of $N$. In this paper, we prove that for eac...

The aim of this note is to show that given a positive integer $n \geq 5$, the positive integral solutions of the diophantine equation $4/n = 1/x + 1/y+1/z$ cannot have solution such that $x$ and $y$ are coprime with $xy < \sqrt{z/2}$. The proof uses the continued fraction expansion of $4/n$.

In this article, for a certain subset $\mathcal{X}$ of the extended set of rational numbers, we introduce the notion of {\it best $\mathcal{X}$-approximations} of a real number. The notion of best $\mathcal{X}$-approximation is analogous to that of best rational approximation. We explore these approximations with the help of $\mathcal{F}_{p^l}$-continued fractions, where $p$ is a prime and $l\in\mathbb{N}$, we show that the convergents of the $\mathcal{F}_{p^l}$-continued fra...

Continued fractions in the field of $p$--adic numbers have been recently studied by several authors. It is known that the real continued fraction of a positive quadratic irrational is eventually periodic (Lagrange's Theorem). It is still not known if a $p$--adic continued fraction algorithm exists that shares a similar property. In this paper we modify and improve one of Browkin's algorithms. This algorithm is considered one of the best at the present time. Our new algorithm ...

Let $\mathbb{Z}_p$ be the ring of $p$-adic integers and $a_n(x)$ be the $n$-th digit of Schneider's $p$-adic continued fraction of $x\in p\mathbb{Z}_p$. We study the growth rate of the digits $\{a_n(x)\}_{n\geq1}$ from the viewpoint of multifractal analysis. The Hausdorff dimension of the set \[E_{\sup}(\psi)=\Big\{x\in p\mathbb{Z}_p:\ \limsup\limits_{n\to\infty}\frac{a_n(x)}{\psi(n)}=1\Big\}\] is completely determined for any $\psi:\mathbb{N}\to\mathbb{R}^{+}$ satisfying $\p...

Let $|| \cdot ||$ denote the distance to the nearest integer and, for a prime number $p$, let $| \cdot |_p$ denote the $p$-adic absolute value. In 2004, de Mathan and Teuli\'e asked whether $\inf_{q \ge 1} \, q \cdot || q \alpha || \cdot | q |_p = 0$ holds for every badly approximable real number $\alpha$ and every prime number $p$. Among other results, we establish that, if the complexity of the sequence of partial quotients of a real number $\alpha$ grows too rapidly or too...