JARNIK'S THEOREM WITHOUT THE MONOTONICITY ON THE APPROXIMATING FUNCTION

摘要

Let psi : N -> R->= 0 be a non-negative function such that psi(q) -> 0 as q -> infinity. The well-known Jarnik-Besicovtich theorem concerns the Hausdorff dimension of the set of psi- approximable numbers. In this paper, we give an alternative but short proof of the Jarnik-Besicovitch theorem for approximating functions with no monotonicity. The main tool is the appropriate usage of the mass transference principle of Beresnevich-Velani [A mass transference principle and the Duffin-Schaeffer conjecture for Hausdorff measures, Ann. of Math. (2) 164(3) (2006) 971-992].