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On the $$\beta$$-expansions of real numbers. (English) Zbl 0099.28103

##### MSC:
 11K55 Metric theory of other algorithms and expansions; measure and Hausdorff dimension
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##### References:
 [1] A. Rényi, Representations for real numbers and their ergodic properties,Acta Math. Acad. Sci. Hung.,8 (1957), pp. 477–493. · Zbl 0079.08901 · doi:10.1007/BF02020331 [2] P. R. Halmos,Lectures on ergodic theory, Mathematical Society of Japan (1956). · Zbl 0073.09302 [3] F. Riesz etB. Sz.-Nagy,Leçons d’analyse fonctionnelle (Budapest, 1952). [4] P. R. Halmos,Measure theory (New York, 1950). [5] M. E. Munroe,Introduction to measure and integration (Cambridge, 1953). · Zbl 0050.05603 [6] J. W. S. Cassels,An introduction to diophantine approximations, Cambridge Tracts (1957). · Zbl 0077.04801 [7] G. D. Birkhoff, Proof of the ergodic theorem,Proc. Nat. Acad. Sci. USA,17 (1931), pp. 162–166.
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