Beyond Expansion II: Low-Lying Fundamental Geodesics
  A closed geodesic on the modular surface is "low-lying" if it does not travel "high" into the cusp. It is "fundamental" if it corresponds to an element in the class group of a real quadratic field. We prove the existence of infinitely many low-lying fundamental geodesics, answering a question of Einsiedler-Lindenstrauss-Michel-Venkatesh. 
Jean Bourgain and Alex Kontorovich
math.NT math.DS