Beyond Expansion III: Reciprocal Geodesics
  We prove the existence of infinitely many low-lying and fundamental closed geodesics on the modular surface which are reciprocal, that is, invariant under time reversal. The method combines ideas from Parts I and II of this series, namely the dispersion method in bilinear forms, as applied to thin semigroups coming from restricted continued fractions. 
Jean Bourgain and Alex Kontorovich
math.NT