Liminf Sets in Simultaneous Diophantine Approximation

Abstract

Let 𝒬 be an infinite set of positive integers. Denote by W τ,n * (𝒬) the set of n–tuples of real numbers simultaneously τ–well approximable by infinitely many rationals with denominators in 𝒬 but by only finitely many rationals with denominators in the complement of 𝒬. The Hausdorff dimension of the liminf set W τ,n * (𝒬) is determined when n≥1 and τ>2+1/n. A p–adic analogue of the problem is also studied.