arXiv subjects

Gerardo González Robert

Publications and source records attributed to Gerardo González Robert.

13 recordsLinked to original sources

arXiv2026

Measure theoretic properties of large products of consecutive partial quotients

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 $φ:\mathbb{N}\to\mathbb{R}_{\geq 2}$, we determine the Lebesgue measure of the set $\mathcal{F}_{\ell}(φ)$ of irrational numbers $x$ whose regular continued fraction $x~=~[a_1(x),a_2(x),\ldots]$ is such that, for infinitely many $n\in\mathbb{N}$, there are two numbers $1\leq j<k \leq n$ satisfying \[ a_{k}(x)\cdots a_{k+\ell-1}(x) \geq φ(n), \; a_{j}(x)\cdots a_{j+\ell-1}(x) \geq φ(n). \] This result generalizes previous work by Tan and Zhou (Nonlinearity, 2024). A consequence of our result is that the strong law of large numbers for products of $\ell$ consecutive partial quotients is impossible even if the block with the largest product is removed. We also compute the Hausdorff dimension of $\mathcal{F}_3(φ)$.

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arXiv2025

Continued fractions with large prime partial quotients

We determine the Lebesgue measure and Hausdorff dimension of various sets of real numbers with infinitely many partial quotients that are both large and prime, thus extending the well-known theorems by Łuczak (1997) and Huang-Wu-Xu (2020). To this end, we obtain new asymptotics on the tail end of the almost prime zeta function. Our results include some recent work by Schindler-Zweim{ü}ller (2023).

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arXiv2025

Approximation by uniformly distributed sequences

We consider approximation properties of real points by uniformly distributed sequences. Under some assumptions on the approximation functions, we prove a Khintchine-type $0$-$1$ dichotomy law. We establish a new connection between uniform distribution and the ubiquity property. Namely, we show that a bound on the discrepancy of the sequence implies the ubiquity property, which helps to obtain divergence results. We further obtain Hausdorff dimension results for weighted sets. The key tools in proving these results are the weighted ubiquitous systems and weighted mass transference principle introduced recently by Kleinbock \& Wang, and Wang \& Wu respectively.

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arXiv2025

Weighted approximation for limsup sets

Theorems of Khintchine, Groshev, Jarník, and Besicovitch in Diophantine approximation are fundamental results on the metric properties of $Ψ$-well approximable sets. These foundational results have since been generalised to the framework of weighted Diophantine approximation for systems of real linear forms (matrices). In this article, we prove analogues of these weighted results in a range of settings including the $p$-adics (Theorems 7 and 8), complex numbers (Theorems 9 and 10), quaternions (Theorems 11 and 12), and formal power series (Theorems 13 and 14). The key tools in proving the main parts of these results are the weighted ubiquitous systems and weighted mass transference principle introduced recently by Kleinbock--Wang [Adv. Math. (2023)] and Wang--Wu [Math. Ann. (2021)].

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arXiv2025

Transcendence and normality of complex numbers via Hurwitz continued fractions

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. The resulting space of sequences of Gaussian integers $Ω$ is not closed. By means of an algorithm, we show that $Ω$ contains a natural subset whose closure $\overline{\mathsf{R}}$ encodes continued fraction expansions of complex numbers which are not Gaussian rationals. We prove that $(\overline{\mathsf{R}}, σ)$ is a subshift with a feeble specification property. As an application, we determine the rank in the Borel hierarchy of the set of Hurwitz normal numbers with respect to the complex Gauss measure. We also construct a family of complex transcendental numbers with bounded partial quotients.

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arXiv2023

Restricted slowly growing digits for infinite iterated function systems

For an infinite iterated function system $\mathbf{f}$ on $[0,1]$ with an attractor $Λ(\mathbf{f})$ and for an infinite subset $D\subseteq \mathbb{N}$, consider the set \[ \mathbb E(\mathbf{f},D)= \{ x \in Λ(\mathbf{f}): a_n(x)\in D \text{ for all }n\in\mathbb N \text{ and }\lim_{n\to\infty} a_n=\infty\}. \] For a function $φ:\mathbb{N}\to [\min D, \infty)$ such that $φ(n)\to\infty$ as $n\to\infty$, we compute the Hausdorff dimension of the set $$ S(\mathbf{f},D,φ) = \left\{ x\in \E(\mathbf{f},D) : a_n(x)\leq φ(n) \text{ for all } n\in\mathbb N \right\}. $$ We prove that the Hausdorff dimension stays the same no matter how slowly the function $φ$ grows. One of the consequences of our result is the recent work of Takahasi (2023), which only dealt with regular continued fraction expansions. We further extend our result to slowly growing products of (not necessarily consecutive) digits.

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arXiv2023

Complex numbers with a prescribed order of approximation and Zaremba's conjecture

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 $τ\geq 2$ 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 integers) equal to $τ$. One of the key ingredients in our construction is the `Folding Lemma' applied to Hurwitz continued fractions. This motivates a Hurwitz continued fraction analogue of the well-known Zaremba's conjecture. We prove several results in support of this conjecture.

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arXiv2023

Metrical properties of weighted products of consecutive Lüroth digits

The Lüroth 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 $m\in \mathbb{N}$, $\mathbf{t}=(t_0,\ldots, t_{m-1})\in\mathbb{R}_{>0}^{m-1}$ and any function $Ψ:\mathbb{N}\to (1,\infty)$, define \[ \mathcal{E}_{\mathbf{t}}(Ψ)\colon= \left\{ x\in (0,1]: d_n^{t_0} \cdots d_{n+m}^{t_{m-1}}\geq Ψ(n) \text{ for infinitely many} \ n \in\mathbb{N} \right\}. \] We establish a Lebesgue measure dichotomy statement (a zero-one law) for $\mathcal{E}_{\mathbf{t}}(Ψ)$ under a natural non-removable condition $\liminf_{n\to\infty} Ψ(n)>~1$. Let $B$ be given by \[ \log B \colon= \liminf_{n\to\infty} \frac{\log(Ψ(n))}{n}. \] For any $m\in\mathbb{N}$, we compute the Hausdorff dimension of $\mathcal{E}_{\mathbf{t}}(Ψ)$ when either $B=1$ or $B=\infty$. We also compute the Hausdorff dimension of $\mathcal{E}_{\mathbf{t}}(Ψ)$ when $1<B< \infty$ for $m=2$.

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arXiv2022

Chaotic sets and Hausdorff dimension for Lüroth expansions

We provide new similarities between regular continued fractions and Lüroth series in terms of topological dynamics and Hausdorff dimension. In particular, we establish a complete analogue for the Lüroth transformation of results by W. Liu, B. Li and W. Liu, S. Wang on the distal, asymptotic and Li-Yorke pairs for the Gauss map.

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arXiv2021

A Complex Borel-Bernstein Theorem

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. As a corollary, we obtain a complex version of Khinchin's theorem on Diophantine approximation.

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arXiv2021

Hausdorff dimension of sets of numbers with large Lüroth elements

Lüroth 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 their growth. Although different, regular continued fractions and Lüroth series share several properties. In this paper, we explore one similarity by estimating the Hausdorff dimension of subsets of real numbers whose Lüroth expansion grows at a definite rate. This is an extension of a result of Y. Sun and J. Wu to the context of Lüroth series. It was recently shown by Y. Feng, B. Tan, and Q.-L. Zhou that the lower bound in our main theorem is actually an equality.

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arXiv2020

Good's Theorem for Hurwitz Continued Fractions

Good's Theorem for regular continued fraction states that the set of real numbers $[a_0;a_1,a_2,\ldots]$ such that $\displaystyle\lim_{n\to\infty} a_n=\infty$ has Hausdorff dimension $\tfrac{1}{2}$. We show an analogous result for the complex plane and Hurwitz Continued Fractions. The set of complex numbers whose Hurwitz Continued fraction $[a_0;a_1,a_2,\ldots]$ satisfies $\displaystyle\lim_{n\to\infty} |a_n|=\infty$ has Hausdorff dimension $1$, half of the ambient space's dimension.

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arXiv2013

Generalized Thue-Morse Continued Fractions

The Thue-Morse sequence is generalized to the $TM_m$ sequences and two equivalent definitions are given. This generalization leads to transcendental numbers and has Queffélec's theorem on Thue-Morse continued fractions as a special case. It is also shown that the theorem of Adamczewski and Bugeaud for palindromic continued fractions cannot be applied in this case.

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