Gerardo González Robert

• PhD
• Research Officer at La Trobe University

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 that is not a Gaussian rational. The resulting space of sequences of Gaussian integers $\Omega $ is not closed. Using an iterative proce...

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 $\ell\in \mathbb{N}$, we determine the Lebesgue measure and Hausdorff dimension of the set $\mathcal{F}_{\ell}(\varph...

For an infinite iterated function system f on [0, 1] with an attractor Λ(f) and for an infinite subset D ⊆ N, consider the set E(f , D) = {x ∈ Λ(f) : a n (x) ∈ D for all n ∈ N and lim n→∞ a n = ∞}. For a function φ : N → [min D, ∞) such that φ(n) → ∞ as n → ∞, we compute the Hausdorff dimension of the set S(f , D, φ) = {x ∈ E(f , D) : a n (x) ≤ φ(n...

Given b = −A ± i with A being a positive integer, we can represent any complex number as a power series in b with coefficients in A = {0, 1,. .. , A 2 }. We prove that, for any real τ ≥ 2 and any non-empty proper subset J(b) of A, there are uncountably many complex numbers (including transcendental numbers) that can be expressed as a power series i...

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 weighted Diophantine approximation for systems of real linear forms (matrices). In this article, we prov...

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 continued fractions and, as a consequence, we contribute significantly in establishing the metrical theory...

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 $m\in \mathbb{N}$, $\mathbf{t}=(t_0,\ldots, t_{m-1})\in\mathbb{R}_{>0}^{m-1}$ and any function $\Psi:\mathbb{N}\to (1,\inft...

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 [18] and W. Liu, S. Wang [19] on the distal, asymptotic and Li-Yorke pairs for the Gauss map.

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.

Regular continued fraction expansions and Lüroth series of real numbers within the unit interval share several properties, although they are generated by different dynamical systems. Our research provides new similarities between both sets of expansions in terms of topological dynamics and Hausdorff dimension. In particular, we establish a complete...

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 real numbers according to their growth. Although different, regular continued fractions and Lüroth series share several proper...

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.

L\"uroth series, like regular continued fractions, provide an interesting identification of irrational numbers with infinite sequence 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 L\"uroth series share several pr...

Good’s Theorem for regular continued fraction states that the set of real numbers [a0; a1,a2,…] such that limn→∞an = ∞ has Hausdorff dimension 1 2. We show an analogous result for the complex plane and Hurwitz Continued Fractions: the set of complex numbers whose Hurwitz Continued fraction [a0; a1,a2,…] satisfies limn→∞|an| = ∞ has Hausdorff dimens...

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_...

Adolf Hurwitz proposed in 1887 a continued fraction algorithm for complex numbers: Hurwitz continued fractions (HCF). Among other similarities between HCF and regular continued fractions, quadratic irrational numbers over $\mathbb{Q}(i)$ are precisely those with periodic HCF expansions. In this paper, we give some necessary as well as some sufficie...