Gero's Math Website

Mathematical Texts

Research

Published mathematical communication

Notes and Expository Articles

Research

I am interested in metric Diophantine approximation, especially regular continued fractions, Lüroth Series and mostly Hurwitz Continued Fractions. Recently, I have also been working on dynamical aspects of such representations too.

Publications
The published versions are slightly different from the last arxiv versions. If you want a copy of the published version, don't hesitate to contact me.

with R. Alcaraz Barrera, "Chaotic sets and Hausdorff dimension for Lüroth expansions". Journal of Mathematical Analysis and its Applications. Volume 514, Issue 2, 15 October 2022, 126324 (journal, arxiv)
A complex Borel-Bernstein Theorem. Bol. Soc. Mat. Mex. 28, 7 (2022). (journal, read only, arxiv)
with Aubin Arroyo Camacho. Hausdorff dimension of Sets of Numbers with Large Lüroth Elements. Integers 21 (2021), Paper No. A71, 20 pp. 11K55. (journal)
Good's Theorem for Hurwitz Continued Fractions. Int. Journal of Number Theory. (2020) Volume No.16, Issue No. 07. (arxiv, journal)
Purely Periodic and Transcendental Continued Fractions. Acta Arith. 194 (2020), no. 3, 241--265. (pdf, journal)

Preprints

with Mumtaz Hussain and Adam Brown-Sarre. Measure theoretic properties of large products of consecutive partial quotients (arxiv)
with M. Hussain, N. Shulga, and H. Takahasi. Restricted slowly growing digits for infinite iterated function systems (arxiv)
with Felipe García-Ramos and Mumtaz Hussain. Transcendence and Normality of Complex Numbers (arxiv)
with Mumtaz Hussain and Nikita Shulga. Complex numbers with a prescribed order of approximation and Zaremba's conjecture (arxiv)
with Mumtaz Hussain, Nikita Shulga, and Ben Ward, Weighted approximation for limsup sets (arxiv)
with Yann Bugeaud and Mumtaz Hussain, Metrical properties of Hurwitz Continued Fractions (arxiv)
with Mumtaz Hussain and Adam Brown-Sarre. Metrical Properties of Weighted Products of Consecutive Lüroth digits (here).

Work in progress

I am working with Mumtaz Hussain and Johannes Schleischitz on a complex Diophantine approximation project.
I am studying with Simon Kristensen and Mumtaz Hussain expansions on integer base.
I am studying with Nikita Shulga, Mumtaz Hussain, and Zhenliang Zhang expansions on integer base.
Some surprises without continued fractions!

Published mathematical communication

2018

Aproximando a la razón áurea y a otros números. (Spanish) (Eng: Approximating the Golden Ratio and Other Numbers) Laberintos & Infinitos No. 46 (2018), pp. 7-14. México. Published Version. My edition.

Notes and Expository Articles

A Failed Complex Continued Fraction. (Letter to Maxim Kirsebom) (English, 2 p.) In this note, it is shown that the straightforward generalization of the regular continued fraction to the complex plane does not work. Despite being a well known result, I was not we aware of any detailed explanation in the literature. (pdf)
Numeros Mal Aproximables (English: Badly Approximable Numbers)
(Spanish, 8 p.) Signed as "Gero". A basic result in Diophantine approximation is that badly approximable numbers are a Lebesgue null set. In this note, I discuss the proof of the previous result given in Khinchine's book "Continued Fractions". (pdf)

Theses

PhD Thesis (English). Complex Continued Fractions. Theoretical Aspects of Hurwitz's Algorithm (167 p.) (2018). It is a survey on the theory of Hurwitz continued fractions (HCF) including original results. It starts considering HCF within the complex continued fraction framework of S.G. Dani and A. Nogueira. Afterwards, it explores algebraic properties, ergodic theoretical aspects, and finishes exploring their fractal geometrical aspects. (pdf)
Master's Thesis (Spanish). Aproximación Diofantina. Trascendencia y algebraicidad de números (126 p.) (2015) Description: The thesis contains a detailed proof of Roth's Theorem on Diophantine Approximation, of Schmidt's Subpace Theorem, and an application to automatic continued fractions (mostly based on works by J.W. Cassels, W. Schmidt, and Y. Bugeaud). (pdf)
Bachelor's Thesis (Spanish). Fracciones encadenadas. (139 p.) (2011) Description: It is a survey of the elementary theory of regular continued fractions. Some measure theoretic aspects as well as representation of transcendental numbers through palindromic continued fractions are also treated (mostly based on works by A. Khinchin, and B. Adamczewski and Y. Bugeaud). (pdf)

Talks

Summer School. 4-8/07/2022. The website of the summer school «Geometría Prohibida» (Forbidden Geometry) is here. San Luis Potosí, Mexico.
10/2021. My tals on the National Congress of the Mexican Mathematical Society are here (in Spanish): "Complex continued fractions" and "Lüroth Series".
11/03/2021. Title: "Chaos in Lüroth series". Seminario Dinámico Potosino, Universidad Autónoma de San Luis Potosí (slides in Spanish). Joint work with Rafael Alcaraz Barrera
16/02/2021. Title: "Good's Theorem for Hurwitz continued Fractions". One World Numeration Seminar. (Slides, seminar website, video)
23/10/2020. Title: Números reales con series de Lüroth con crecimiento rápido (English: Real numbers with Lüroth series of fast growth.) (sildes in Spanish, video). Congreso Nacional de la Sociedad Matemática Mexicana.

Classroom Notes

2021.
I am currently writing classroom notes for the courses "Mathematical Analysis II" and "Mathematical Analysis III". These notes are available at the courses websites.
2020.
I wrote the following notes for the class "Mathematical Analysis I" at the Facultad de Ciencias, UNAM. Comments are welcome. All of them are in Spanish.

Two uncountable spaces pdf
Metric spaces pdf
Examples of metric spaces pdf
Compact sets pdf
Sequences pdf

2018

Examples of Sequences. (Course: Mathematical Analysis, Aarhus University) In this note, the limes inferior and superior of three different sequences are discussed. The notation is standard and was established in the course's lecture notes. (pdf)

Past courses