We study the small scale distribution of the L^2-mass of eigenfunctions of the Laplacian on the the two-dimensional flat torus. Given an orthonormal basis of eigenfunctions, Lester and Rudnick showed the existence of a density one subsequence whose L^2-mass equidistributes more-or-less down to the Planck scale. We give a more precise version of their result showing equidistribution holds down to a small power of log above Planck scale, and also showing that the L^2-mass fails to equidistribute at a slightly smaller power of log above the Planck scale. This article rests on a number of results about the proximity of lattice points on circles, much of it based on foundational work of Javier Cilleruelo.

In this Monthly note, we use Van der Waerden's Theorem and Fermat's Theorem on four squares in an arithmetic progression to prove that there are infinitely many primes.

We show that smooth-supported multiplicative functions \( f\) are well-distributed in arithmetic progressions \( a_1a_2^{-1} \pmod q\) on average over moduli \( q\leq x^{3/5-\varepsilon}\) with \( (q,a_1a_2)=1\) . This extends our results in Bombieri-Vinogradov for multiplicative functions, and beyond the \( x^{1/2}\)-barrier (with Fernando Shao).

There are many different ways to prove that there are infinitely many primes. I will highlight a few of my favourites, selected so as to involve a rich variety of mathematical ideas.