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ON THE DIOPHANTINE PROPERTIES OF λ-EXPANSIONS

Part of: Diophantine approximation, transcendental number theory Classical measure theory

Published online by Cambridge University Press:  05 December 2012

Tomas Persson  and
Henry W. J. Reeve
Tomas Persson
Affiliation:
Centre for Mathematical Sciences, Lund University, Box 118, 22100 Lund, Sweden (email: tomasp@maths.lth.se)
Henry W. J. Reeve
Affiliation:
Department of Mathematics, The University of Bristol, University Walk, Clifton, Bristol, BS8 1TW, U.K. (email: henrywjreeve@gmail.com)
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Abstract

For and α, we consider sets of numbers x such that for infinitely many n, x is 2−αn-close to some ∑ ni=1ωiλi, where ωi∈{0,1}. These sets are in Falconer’s intersection classes for Hausdorff dimension s for some s such that −(1/α)(log λ /log   2 )≤s≤1/α. We show that for almost all , the upper bound of s is optimal, but for a countable infinity of values of λ the lower bound is the best possible result.

MSC classification

Secondary: 11J83: Metric theory 28A78: Hausdorff and packing measures
Type
Research Article
Information
Mathematika , Volume 59 , Issue 1 , January 2013 , pp. 65 - 86
Copyright
Copyright © University College London 2012

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