A note on the Hausdorff dimension of some liminf sets appearing in simultaneous Diophantine approximation

TL;DR

This paper investigates the Hausdorff dimension of certain liminf sets in simultaneous Diophantine approximation, providing lower bounds under specific divisibility conditions for sets of denominators.

Contribution

It establishes a non-trivial lower bound for the Hausdorff dimension of liminf sets in higher dimensions when the approximation exponent exceeds a threshold, under divisibility assumptions.

Findings

Lower bounds for Hausdorff dimension of liminf sets in higher dimensions.

Discussion on exact Hausdorff dimension and one-dimensional cases.

Abstract

Let Q be an infinite set of positive integers. Denote by W_{\tau, n}(Q) (resp. W_{\tau, n}) the set of points in dimension n simultaneously \tau--approximable by infinitely many rationals with denominators in Q (resp. in N*). A non--trivial lower bound for the Hausdorff dimension of the liminf set W_{\tau, n}\W_{\tau, n}(Q) is established when n>1 and \tau >1+1/(n-1) in the case where the set Q satisfies some divisibility properties. The computation of the actual value of this Hausdorff dimension as well as the one--dimensional analogue of the problem are also discussed.