Akshat Mudgal

Counting solutions to the quadratic determinant equation

Let h, N be positive integers. In this talk, we will count solutions to the equation x_1 x_2 - x_3 x_4 = h, with |x_i| < N. In particular, we would like to prove an asymptotic formula with power saving for many choices of h. This was done by Afifurrahman when h < N^{2 - o(1)}, with better error terms given by Ganguly–Guria when h < N^{1/3}. In joint work with Chapman, we obtain asymptotic formulae for a much wider range of h with stronger error terms using significantly simpler methods.