I am a third-year PhD student under the supervision of Dr. Simon Myerson and Dr. Sam Chow.

E-mail address: Nuno.Arala-Santos(at)warwick.ac.uk.

Interests

Number Theory, especially interactions between Analytic Number Theory and Diophantine Geometry.

Background

I obtained my Bachelor's degree from the University of Porto in 2018. I then did my Master's at the University of Bonn, which I finished in 2020. I wrote my Master's Thesis under the supervision of Prof. Valentin Blomer on the estimation of Representation Numbers of Quadratic Forms using a variant of the Hardy-Littlewood circle method. (For more details see below, under "Writing".)

For more information on my background see my CV.

Papers and Preprints

1. A maximal extension of the Bloom-Maynard bound for sets with no square differences (2023) (arXiv, .pdf), submitted
2. Lines of polynomials with alternating Galois group (2022), (arXiv, .pdf), submitted
3. On the density of a set of primes associated to an Elliptic Curve (2021), (arXiv, .pdf), Journal of Number Theory 243 (2023) 241-255.

Miscellaneous writing

Below are some math-related texts I wrote over the years, for various purposes. Several of them are written in Portuguese. I intend to translate some of those to English at some point in the future.

• (2014, in Portuguese) Three handouts on Euclidean Geometry topics, intended for Olympiad students: Geometry for Beginners, Random configurations and Brute Force. These are joint work with Miguel Moreira.
• (2015, in Portuguese) A short handout on the "Lifting the Exponent" Lemma, intended for Olympiad students: Lifting the Exponent.
• (2019, in Portuguese) Affine and Projective Geometry. This text was loosely inspired in a course I took in the final year of my Bachelor's.
• (2020) A short text on the classification of Quaternion Algebras over Local Fields.
• (2020) My Master's Thesis on Representation Numbers of Quadratic Forms.