Given a positive definite integral binary quadratic form, it is a classical problem in number theory to count the integers that are represented by this form. A modern treatment was given in 2006 by Valentin Blomer and Andrew Granville.
This talk will present a way of extending a theorem of Blomer and Granville to obtain estimates for counting proper representations uniform in the (possibly non-fundamental) discriminant. Subsequently, I will give a sketch of how these estimates were used by Jean Bourgain and Elena Fuchs (2011) in proving the positive density conjecture for Apollonian circle packings.
Links:
[1] https://www.mpim-bonn.mpg.de/de/taxonomy/term/39
[2] https://www.mpim-bonn.mpg.de/de/node/3444
[3] https://www.mpim-bonn.mpg.de/de/node/246