# Research

My research interests are wide in Number Theory, Diophantine Approximation and Diophantine Equations, including in particular Modular Forms, Continued Fractions, Binary Quadratic Forms, Thue Equations and Thue Inequalities, Metric Number Theory.

## Publications

Thue equations that simultaneously fail the Hasse principle, *submitted*.

Thue inequalities with few coefficients, *International Mathematics Research Notices, to appear*.

Representation of integers by sparse binary forms (with S. Akhtari),
*Transactions of the American Mathematical Society, to appear*.

Values of modular functions at real quadratics and conjectures of Kaneko (with O. Imamoglu),
*Mathematische Annalen*, 377(1) (2020), 249-266.

Cycle integrals of modular functions, Markov geodesics and a conjecture of Kaneko (with O. Imamoglu)
*Algebra & Number Theory* 13-4 (2019), 943-962.

Metric number theory of Fourier coefficients of modular forms
*Proceedings of the American Mathematical Society* 147-07 (2018), 2835-2845.

A note on badly approximable forms on
manifolds (with N. Moshchevitin and N. Stepanova) *Mathematika* 63 (2) (2017) 587-601.

Badly approximable points in twisted Diophantine approximation and Hausdorff dimension (with N. Moshchevitin) *Acta Arithmetica* 177 (4) (2017) 301-314.

On the Mixed Littlewood Conjecture and continued fractions in quadratic fields (with E. Zorin) *Journal of Number Theory* 162 (2016) 1-10.

On a theorem of Serret on continued fractions *Revista de la Real Academia
de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas*, 110 (2) (2016) 379-384.

Galois action on special theta values *Journal de Théorie des Nombres de
Bordeaux*, 28 (2) (2016) 347-360

Meromorphic analogues of modular forms generating the kernel of Shimura's
lift *Mathematical Research Letters*, 22 (2) (2015) 337-352.

From quadratic polynomials and continued fractions to modular forms *Journal
of Number Theory* 147 (2015) 24-43.

## Talks

The following is a list of seminars and meetings where I have given research talks.

**International Seminar on Automorphic Forms**, via Zoom. April 2020. Periods of modular functions and Diophantine approximation (slides: https://www2.mathematik.tu-darmstadt.de/~international_seminar/pdfs/handout_bengoechea.pdf)

**Seminari de Teoria de Nombres de Barcelona**, Barcelona. February 2020.

**Seminaire tournant de theorie des nombres**, Institut Fourier Grenoble. April 2019.

**Number Theory Seminar**, University of Luxembourg. December 2018.

**N^3-Days IX (Nordic Number Theory Network)**, Lund University. 30 November - 1 December 2018.

**Seminar of Algebraic Geometry and Number Theory**, EPFL Lausanne. November 2018.

**Elementare und Analytische Zahlentheorie (ELAZ)**, Max Planck Institute for Mathematics. September 3-7, 2018.

**Number Theory seminar**, Max Planck Institute for Mathematics. May 2018.

** Seminar in Algebraic Geometry and Number Theory**, Chalmers University of Technology and University of Gothenburg. November 2017.

** Seminari de Teoria de Nombres de Barcelona (STNB)**, Barcelona. January 2017.

** Geometric and Analytic Number Theory meeting**, ETH Zurich. 12-15 September 2016.

** Number theory seminar**, University of Manchester. April 2016.

** Pure Mathematics Seminar**, University of Reading. March 2016.

** Heilbronn Seminar**, Bristol. March 2016.

** Number Theory Seminar**, University of Warwick. March 2016.

** Pure Mathematics Seminar**, Exeter University. February 2016.

** Number Theory and Geometry Seminar**, University of Nottingham. February 2016.

** Séminaire de théorie des nombres**, Institut Fourier, Grenoble. January 2016.

** Seminari de Teoria de Nombres de Barcelona (STNB)**, Barcelona. January 2016.

** Conference: `Arithmetic geometry: explicit methods and applications'**, Moscow. 7-11 December 2015.

** Séminaire d'Algèbre et de Théorie des Nombres**, EPFL, Lausanne. December 2015.

** Séminaire tournant de théorie de nombres**, Lyon. October 2015.

** Arithmetic Study Group**, Durham. December 2014.

** Newton Institute for Mathematical Sciences Seminar**, Cambridge. July 2014.

** Number Theory Seminar**, Cologne. June 2014.

** Logic and Number Theory Seminar**, Oxford. February 2014.

** Algebra, geometry and number theory seminar**, Leiden. May 2013.

** Heilbronn Seminar**, Bristol. May 2013.

** Number Theory and Physics Seminar of Bonn-Koeln**, Max Planck Institute for Mathematics, Bonn. May 2013.

** Tercera jornada SCM de jóvenes matemáticos investigadores**, Barcelona. October 2012.

** Study group: Cohomologie des groupes**, Institut de Mathématiques de Jussieu,
Paris. February 2010.

** Seminari de Teoria de Nombres de Barcelona (STNB)**, Barcelona. January 2008.