Effective Counting of the $3$-dimensional Markov Spectrum

Let $O3$ be the set of non-degenerate indefinite real quadratic forms in $3$ variables. For any $Q\in O3$ we define $m(Q)=inf\{|Q(w)|: w\in Z3 \{0\}\}$. Let $det(Q)$ be the determinant of $Q$. The set $M3={m(Q)3/det Q: Q\in O3}$ is called the $3$-dimensional Markov spectrum. An early result of Cassels-Swinnerton-Dyer combined with Margulis’ proof of the Oppenheim conjecture asserts that, for every $a0$ the set $M3\cap (a, ∞)$ is finite. In this lecture we shall give an effective counting result: there is a constant $C0$ independent of $a$, such that
$|M3\cap(a, ∞)|C a^{-26}$.

This is a joint work with Prof. Margulis, and our approach is based on dynamics on homogeneous spaces.