JARNÍK'S THEOREM WITHOUT THE MONOTONICITY ON THE APPROXIMATING FUNCTION.

Let ψ : ℕ → ℝ ≥ 0 be a non-negative function such that ψ (q) → 0 as q → ∞. The well-known Jarník–Besicovtich theorem concerns the Hausdorff dimension of the set of ψ - approximable numbers. In this paper, we give an alternative but short proof of the Jarník–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]. [ABSTRACT FROM AUTHOR]