Hausdorff measure and linear forms

Detta Dickinson; Sanju Velani

doi:10.1515/crll.1997.490.1

Hausdorff measure and linear forms

Detta Dickinson, Sanju Velani

Mathematics

Research output: Contribution to journal › Article › peer-review

Abstract

Given a dimension function f we prove that the Hausdorff measure H-f of the set W(m, n; psi) of 'well approximable' linear forms is determined by the convergence or divergence of the sum

(r=1)Sigma(infinity) f(psi(r))psi(r)(-(m-1)nrm+n-1).

This is a Hausdorff measure analogue of the classical Khintchine-Groshev Theorem where the mn-dimensional Lebesgue measure of W(m, n; psi) is determined by the convergence or divergence of an mn-volume sum. Our results show that there is no dimension function for which H-f(W(m, n; psi)) is positive and finite.

Original language	English
Pages (from-to)	1-36
Number of pages	36
Journal	Journal fur die reine und angewandte Mathematik (Crelle's Journal)
Volume	490
Issue number	490
DOIs	https://doi.org/10.1515/crll.1997.490.1
Publication status	Published - 1997

Keywords

DIMENSION

Access to Document

10.1515/crll.1997.490.1

Cite this

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AU - Dickinson, Detta

AU - Velani, Sanju

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AB - Given a dimension function f we prove that the Hausdorff measure H-f of the set W(m, n; psi) of 'well approximable' linear forms is determined by the convergence or divergence of the sum(r=1)Sigma(infinity) f(psi(r))psi(r)(-(m-1)nrm+n-1).This is a Hausdorff measure analogue of the classical Khintchine-Groshev Theorem where the mn-dimensional Lebesgue measure of W(m, n; psi) is determined by the convergence or divergence of an mn-volume sum. Our results show that there is no dimension function for which H-f(W(m, n; psi)) is positive and finite.

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