Liminf Sets in Simultaneous Diophantine Approximation

TL;DR

This paper investigates the Hausdorff dimension of liminf sets in simultaneous Diophantine approximation for specific denominator sets and extends the analysis to a p-adic context, providing new dimension formulas under certain conditions.

Contribution

It computes the Hausdorff dimension of the liminf set W(Q) for tau > 2 + 1/n and introduces a p-adic analogue of the problem.

Findings

Dimension formula for W(Q) when tau > 2 + 1/n

Extension of results to p-adic approximation

New insights into liminf sets in Diophantine approximation

Abstract

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