Sets of exact 'logarithmic' order in the theory of Diophantine approximation

Victor Beresnevich; Detta Dickinson; Sanju Velani

doi:10.1007/s002080100225

Sets of exact 'logarithmic' order in the theory of Diophantine approximation

Victor Beresnevich, Detta Dickinson, Sanju Velani

Mathematics

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

Abstract

For each real number , let denote the set of real numbers with exact order . A theorem of Güting states that for the Hausdorff dimension of is equal to . In this note we introduce the notion of exact t–logarithmic order which refines the usual definition of exact order. Our main result for the associated refined sets generalizes Güting's result to linear forms and moreover determines the Hausdorff measure at the critical exponent. In fact, the sets are shown to satisfy delicate zero-infinity laws with respect to Lebesgue and Hausdorff measures. These laws are reminiscent of those satisfied by the classical set of well approximable real numbers, for example as demonstrated by Khintchine's theorem.

Original language	English
Pages (from-to)	253-273
Number of pages	21
Journal	Math Annalen
Volume	321
Issue number	2
DOIs	https://doi.org/10.1007/s002080100225
Publication status	Published - Oct 2001

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10.1007/s002080100225

Cite this

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abstract = "For each real number , let denote the set of real numbers with exact order . A theorem of G{\"u}ting states that for the Hausdorff dimension of is equal to . In this note we introduce the notion of exact t–logarithmic order which refines the usual definition of exact order. Our main result for the associated refined sets generalizes G{\"u}ting's result to linear forms and moreover determines the Hausdorff measure at the critical exponent. In fact, the sets are shown to satisfy delicate zero-infinity laws with respect to Lebesgue and Hausdorff measures. These laws are reminiscent of those satisfied by the classical set of well approximable real numbers, for example as demonstrated by Khintchine's theorem. ",

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