An exponentially shrinking problem

The Jarnik-Besicovitch theorem is a fundamental result in metric number theory which concerns the Hausdorff dimension for some limsup sets. We investigate a related problem of estimating the Hausdorff dimension of a liminf set. Let tau > 0 and {q(j)}(j=1) be a sequence of integers. We calculate the Hausdorff dimension of the set Lambda(theta)(d) (tau) = {x is an element of [0, 1)(d) : parallel to q(j)x(i) - theta(i)parallel to < q(j)(-tau) j for all j >= 1, i = 1, 2, center dot center dot center dot, d}, where parallel to center dot parallel to denotes the distance to the nearest integer and theta is an element of[0, 1)(d) is fixed. We also give some heuristics for the Hausdorff dimension of the corresponding multiplicative set M-d(theta)(tau) = {x is an element of[0, 1)(d) : Pi(d)(i=1) : Pi(d)(i-1) parallel to q(j)x(i) - theta(i)parallel to < q(j)(-tau) j >= 1}. (c) 2023 The Author(s). Published by Elsevier Inc.

Automated summary

The Jarnik-Besicovitch theorem is a fundamental result in metric number theory that deals with the Hausdorff dimension of certain limsup sets. This paper investigates a related problem of estimating the Hausdorff dimension of a liminf set and provides calculations and heuristics for the corresponding multiplicative set.