Applications of Homogeneous Dynamics to Quadratic Forms

Abstract

We study the geometry of quadratic forms using equidistribution theorems in homogeneous dynamics. First we study the mean square limit of exponential sums associated to a rational ellipsoid of arbitrary center. We obtain a lower bound for arbitrary center and that lower bound turns out to be the upper bound as well for ellipsoids with the center of certain diophantine type(see theorem 1.0.4). This result generalizes a work of Marklof.

The second topic is the quantitative Oppenheim conjecture for $S$-arithmetic quadratic forms. For an arbitrary open set $I$ in $\mathbb Q_S$, we show that the number of $S$-integral vectors of norm at most $T$, whose values of an irrational quadratic form are $Q$ in $I$, is asymptotically $c(Q, I)T^{n-2}$ as $T$ goes to infinity. This is a generalization of a work of Eskin-Margulis-Mozes for real case.