Zobrazeno 1 - 10
of 73
pro vyhledávání: '"Robert A Gonzalez"'
The theory of uniform approximation of real numbers motivates the study of products of consecutive partial quotients in regular continued fractions. For any non-decreasing positive function $\varphi:\mathbb{N}\to\mathbb{R}_{>0}$ and $\ell\in \mathbb{
Externí odkaz:
http://arxiv.org/abs/2405.10538
For an infinite iterated function system $\mathbf{f}$ on $[0,1]$ with an attractor $\Lambda(\mathbf{f})$ and for an infinite subset $D\subseteq \mathbb{N}$, consider the set \[ \mathbb E(\mathbf{f},D)= \{ x \in \Lambda(\mathbf{f}): a_n(x)\in D \text{
Externí odkaz:
http://arxiv.org/abs/2312.17388
We study the topological, dynamical, and descriptive set theoretic properties of Hurwitz continued fractions. Hurwitz continued fractions associate an infinite sequence of Gaussian integers to every complex number which is not a Gaussian rational. Th
Externí odkaz:
http://arxiv.org/abs/2310.20029
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 $J(b)$ o
Externí odkaz:
http://arxiv.org/abs/2310.11698
Theorems of Khintchine, Groshev, Jarn\'ik, and Besicovitch in Diophantine approximation are fundamental results on the metric properties of $\Psi$-well approximable sets. These foundational results have since been generalised to the framework of weig
Externí odkaz:
http://arxiv.org/abs/2308.16603
We develop the geometry of Hurwitz continued fractions -- a major tool in understanding the approximation properties of complex numbers by ratios of Gaussian integers. We obtain a detailed description of the shift space associated with Hurwitz contin
Externí odkaz:
http://arxiv.org/abs/2306.08254
The L\"uroth expansion of a real number $x\in (0,1]$ is the series \[ x= \frac{1}{d_1} + \frac{1}{d_1(d_1-1)d_2} + \frac{1}{d_1(d_1-1)d_2(d_2-1)d_3} + \cdots, \] with $d_j\in\mathbb{N}_{\geq 2}$ for all $j\in\mathbb{N}$. Given $m\in \mathbb{N}$, $\ma
Externí odkaz:
http://arxiv.org/abs/2306.06886
We provide new similarities between regular continued fractions and L\"uroth series in terms of topological dynamics and Hausdorff dimension. In particular, we establish a complete analogue for the L\"uroth transformation of results by W. Liu, B. Li
Externí odkaz:
http://arxiv.org/abs/2112.04714
Autor:
Robert, Gerardo González
Zero-one laws are a central topic in metric Diophantine approximation. A classical example of such laws is the Borel-Bernstein theorem. In this note, we prove a complex analogue of the Borel-Bernstein theorem for complex Hurwitz continued fractions.
Externí odkaz:
http://arxiv.org/abs/2104.05129
L\"uroth series, like regular continued fractions, provide an interesting identification of real numbers with infinite sequences of integers. These sequences give deep arithmetic and measure-theoretic properties of subsets of numbers according to the
Externí odkaz:
http://arxiv.org/abs/2010.13932