Effective discreteness of the 3dimensional Markov spectrum
Effective discreteness of the 3dimensional Markov spectrum

Han Li, Yale University
Fine Hall 601
Let the set O={nondegenerate, indefinite, real quadratic forms in 3variables with determinant 1}. We define for every form Q in the set O, the Markov minimum m(Q)=min{Q(v): v is a nonzero integral vector in $R^3$}. The set M={m(Q): Q is in O} is called the 3dimensional Markov spectrum. An early result of CasselsSwinnertonDyer combined with Margulis' proof of the Oppenheim conjecture asserts that, for every a>0 {M \intersect (a, \infty)} is a finite set. In this lecture we will show that
#{M \intersect (a, \infty)}