Maxson Lecture Series

Date: April 11, 2023

Time: 4:00PM - 5:00PM

Location: BLOC 117

Speaker: Kannan Soundararajan, Stanford University

Title: Covering integers using quadratic forms

Abstract: How large must \Delta be so that we can cover a substantial proportion of the integers below X using the binary quadratic forms x^2 +dy^2 with d below \Delta? Problems involving representations by binary quadratic forms have a long history, going back to Fermat. The particular problem mentioned here was recently considered by Hansen and Vaughan, and Diao. In ongoing work with Ben Green, we resolve this problem, and identify a sharp phase transition: If \Delta is below (log X)^{log 2-\epsilon} then zero percent of the integers below X are represented, whereas if \Delta is above (log X)^{log 2 +\epsilon} then 100 percent of the integers below X are represented. I will give a gentle introduction to some of the ideas involved.