Sharp bounds for symmetric and asymmetric Diophantine approximation
  In 2004, J.C. Tong found bounds for the approximation quality of a regular continued fraction convergent of a rational number, expressed in bounds for both the previous and next approximation. We sharpen his results with a geometric method and give both sharp upper and lower bounds. We also calculate the asymptotic frequency that these bounds occur. 
Cor Kraaikamp and Ionica Smeets