Effective discreteness of the 3-dimensional Markov spectrum

Effective discreteness of the 3-dimensional Markov spectrum

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Han Li, Yale University
Fine Hall 601

Let the set O={non-degenerate, indefinite, real quadratic forms in 3-variables with determinant 1}. We define for every form Q in the set O, the Markov minimum m(Q)=min{|Q(v)|: v is a non-zero integral vector in $R^3$}. The set M={m(Q): Q is in O} 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 a>0 {M \intersect (a, \infty)} is a finite set. In this lecture we will show that
#{M \intersect (a, \infty)}