Recent research

My recent research is strongly shaped by my teaching -- I often end up working on ideas I develop during my teaching. This is especially true of the following papers 1. and 3 (which are single-author papers). BMath/MMath students who have taken my courses (especially Algebraic Number Theory) will immediately recognise some of the ideas used in these papers.

Here are my three most recent projects (as of July 2026, link not provided for unsubmitted work):

1. Diophantine approximation: I have always been curious about the Dirichlet approximation theorem, whose standard proof is a simple application of the Pigeonhole Principle (see here, for example). Although the proof is simple, it turns out to be the strongest possible result in some sense, by Roth's Theorem. Now, if you restrict the denominators to be rational primes, the result is known as Vinogradov's theorem. What if you further restrict the primes that are allowed in the denominator? For example, suppose you allow only primes of the form x^2+2y^2?  Or in general, primes which are norms of prime ideals in a number field of arbitrary degree. This is what I have proved in a recent preprint (submitted).

2. Sign changes of Kloosterman sums: In a recent joint work with N. Debouzy, S. Ganguly and O. Ramaré, we studied the sign changes of the Kloosterman sums Kl(1,1;c) where c runs through almost primes (not yet submitted).

3. Adelisation: The adèles are ubiquitous in the theory of automorphic forms and representations. However, I feel that they are underexploited in classical Number Theory. So, in an ongoing work, I have been trying to give an adelic, or rather, adelised proof of some classical results.  This is a sieve-theoretic work (not yet submitted).