Dynamical Systems Seminar

It is a well studied problem to estimate the number of matrices \(A\) with integer coefficients, \(\det(A)=n\), and \(\|A\|\le X\), where \(k\ge 2\), \(n\in \mathbb{Z}\), \(X\ge 1\), and \(\|\cdot\|\) is a prescribed norm. For the case of the Euclidean norm, there are well known asymptotic formulas for this quantity, due to work of Selberg, Duke, Rudnick, Sarnak, and others. For the supremum norm, the asymptotic behavior has been less well understood until recently. In this joint talk, which developed out of a SURF project at UH, we will prove an asymptotic formula with error term (uniform in \(n\) and \(X\)) for the number of \(2\times 2\) matrices with fixed non-zero determinant \(n\), and with coefficients bounded in absolute value by \(X\). We will also explain complementary results that show that the error term in our formula is close to best possible in the delicate range where \(X\) is approximately \(n^{1/2+\delta}\).