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ON THE DIOPHANTINE PROPERTIES OF λ-EXPANSIONS

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

[1]Besicovitch, A., Sets of fractional dimension (IV): on rational approximation to real numbers. J. Lond. Math. Soc. 9 (1934), 126–131.CrossRefGoogle Scholar
[2]Dirichlet, L. G. P., Verallgemeinerung eines Satzes aus der Lehre von den Kettenbrüchen nebst einigen Anwendungen auf die Theorie der Zahlen. S. B. Preuss. Akad. Wiss (1842), 93–95.Google Scholar
[3]Erdős, P., On a family of symmetric Bernoulli convolutions. Amer. J. Math. 61 (1939), 974–976.CrossRefGoogle Scholar
[4]Falconer, K., Classes of sets with large intersections. Mathematika 32 (1985), 191–205.CrossRefGoogle Scholar
[5]Falconer, K., Sets with large intersection properties. J. Lond. Math. Soc. 49 (1994), 267–280.CrossRefGoogle Scholar
[6]Falconer, K., Fractal Geometry. Mathematical Foundations and Applications, 2nd edn, John Wiley & Sons (Hoboken, 2003).CrossRefGoogle Scholar
[7]Färm, D. and Persson, T., Dimension and measure of baker-like skew-products of β-transformations. Discrete Contin. Dyn. Syst. Ser. A 32 (2012), 3525–3537.CrossRefGoogle Scholar
[8]Färm, D., Persson, T. and Schmeling, J., Dimension of countable intersections of some sets arising in expansions in non-integer bases. Fund. Math. 209 (2010), 157–176.CrossRefGoogle Scholar
[9]Jarník, V., Diophantischen approximationen und hausdorffsches mass. Mat. Sb. 36 (1929), 371–382; http://mi.mathnet.ru/eng/msb/v36/i3/p371.Google Scholar
[10]Parry, W., On the β-expansion of real numbers. Acta Math. Acad. Sci. Hunga. 11 (1960), 401–416.CrossRefGoogle Scholar
[11]Pesin, Y., Dimension Theory in Dynamical Systems: Contemporary Views and Applications (Chicago Lectures in Mathematics), University of Chicago Press (Chicago, IL, 1997).CrossRefGoogle Scholar
[12]Pollicott, M. and Simon, K., The Hausdorff dimension of expansions with deleted digits. Trans. Amer. Math. Soc. 347 (1995), 967–983.Google Scholar
[13]Rams, M., Packing dimension estimation for exceptional parameters. Israel J. Math. 130 (2002), 125–144.CrossRefGoogle Scholar
[14]Shmerkin, P. and Solomyak, B., Zeros of {−1,0,1} power series and connectedness loci for self-affine sets. Exp. Math. 15(4) (2006), 499–511.CrossRefGoogle Scholar
[15]Simon, K. and Solomyak, B., On the dimension of self-similar sets. Fractals 10(1) (2002), 59–65.CrossRefGoogle Scholar
[16]Solomyak, B., On the random series ∑ ±λ n (an Erdős problem). Ann. of Math. (2) 142(3) (1995), 611–625.CrossRefGoogle Scholar