Mathematical Outputs
Mathematical Outputs
I am interested in metric Diophantine approximation, especially regular continued fractions, integer base expansions, and (complex) Hurwitz Continued Fractions. Recently, some non-number theoretic aspects of dynamical systems and equidistribution mod 1 have caught my attention.
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 Nikita Shulga, Ben Ward, and Mumtaz Hussain, "Approximation by uniformly distributed sequences". Accepted by the Bulletin of the London Mathematical Society. (Journal, arXiv)
with Nikita Shulga, Hiroki Takahasi, and M. Hussain. Restricted slowly growing digits for infinite iterated function systems. Accepted by the Journal of Mathematical Analysis and Applications (Journal OA)
with Yann Bugeaud and M. Hussain. Metrical properties of Hurwitz continued fractions. Advances in Mathematics. (Journal OA)
with Nikita Shulga and M. Hussain. Complex numbers with a prescribed order of approximation and Zaremba's conjecture. Journal of Number Theory (Journal OA)
with F. García-Ramos and M. Hussain. Transcendence and Normality of Complex Numbers. IMRN. Accepted. (Journal OA)
with A. Brown-Sarre and M. Hussain. Metrical Properties of Weighted Products of Consecutive Lüroth digits. Houston J. Math. 49 (2023), no. 4, 861–897. (journal).
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 A. 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 Mumtaz Hussain, Nikita Shulga, and Ben Ward, Weighted approximation for limsup sets (arxiv)
with Lauren White, Ben Ward, and hussain "Continued Fractions with Large Prime Partial Quotients" (arXiv)
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.
I am working with Ben Ward on certain dynamical systems with holes.
Some surprises without continued fractions!
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.
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)
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)
2022.
A first course in functional analysis. 44 Lectures. YouTube Playlist
2021.
A first course in functional analysis. 45 Lectures. YouTube Playlist
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.
2025.
I am writing classroom notes for my courses "Mathematical Analysis" (elementary topology of metric spaces) and "Mathematical Analysis 2" (measure theory). They can be retrieved here: Mathematical Analysis 1, Mathematical Analysis 2.
2024.
The exams and homework of my functional analysis and measure theory courses are now available here (measure theory) and here (functional analysis).
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.
2018
Examples of Sequences. (Course: Mathematical Analysis, Aarhus University) This note discusses the inferior and superior limes of three different sequences. The notation is standard and was established in the course's lecture notes. (pdf)
February 2022 - June 2022. Análisis 3. Facultad de Ciencias, UNAM (Updated version, 2025)
February 2022 - June 2022. Álgebra Lineal, UASLP
September 2021 - February 2022. Análisis Matemático 2, Facultad de Ciencias, UNAM (2022-01) (Updated version, 2025)
February 2021-August 2021. Facultad de Ciencias, UNAM. Análisis Matemático II
February 2021-August 2021. Facultad de Ciencias, UNAM. Análisis Matemático III
September 2020-February 2021. Facultad de Ciencias, UNAM. Análisis Matemático I
September 2020-February 2021. Facultad de Ciencias, UNAM. Análisis Matemático II
January - June 2020. Facultad de Ciencias, UNAM. Mathematical Analysis I
January - May 2020. Universidad Marista. Differential and Integral Calculus II
January - May 2020. Universidad Marista. Differential and Integral Calculus IV
August - December. Facultad de Ciencias, UNAM. Mathematical Analysis I
Summer 2019. ITAM. Analytic Geometry
I like reading novels and short stories. Recently, I started writing some reviews on my new goodreads account.