Integral Hasse principle for Markoff type cubic surfaces

Abstract: We establish new upper bounds on the number of failures of the integral Hasse principle within the family of Markoff type cubic surfaces $x^2+ y^2+ z^2- xyz= a$ with $|a|\leq A$ as $A\to \infty$. Our bound improves upon existing work of Ghosh and Sarnak. As a result, we demonstrate that the integral Hasse principle holds for a density $1$ of surfaces in certain sparse sequences.