Andrew Granville's Home Page

2009 Publications

Close lattice points on circles (with Javier Cilleruelo)
Canadian Journal of Mathematics, (2009), 1214-1238.

We classify the sets of four lattice points that all lie on a short arc of a circle that has its center at the origin; specifically on arcs of length \( tR^{1/3}\) on a circle of radius \( R\), for any given \( t>0\).


Visibility in the plane (with Sukumar Das Adhikari)
Journal of Number Theory, 129 (2009), 2335-2345.

We find the size of the smallest subset of the set of integer lattice points, such that every element of a given rectangular grid is visible from our subset, which answers a question of Paul Erdos.


Pretentiousness in analytic number theory
Journal de Théorie de nombres de Bordeaux, 21 (2009),159--173.

This is a written version of my lecture at the Journée Arithmétique in Edinburgh, in which I describe how, in joint work with K. Soundararajan and Antal Balog, we have developed the notion of "pretentiousness" to help us better understand several key questions in analytic number theory.