Lu- ming SHEN, Yu-yuan ZHOU, Yue-hua LIU

A Note on a Problem of J. Galambos

For any x\in (0,1], letx = \frac{1}{d1} + \frac{a1}{b1} \frac{1}{d2} + ··· + \frac{a1a2 ··· an}{b1b2 ··· bn} \frac{1}{dn+1} + ··· be the Oppenheim series expansion of x. In this paper, we investigate the Hausdorff dimension of the set Bm={x:1

Anahtar Kelimeler:

Oppenheim series expansion; Restricted Oppenheim series expansion; Lüroth series; Hausdorff dimension

A Note on a Problem of J. Galambos

Keywords:

Oppenheim series expansion; Restricted Oppenheim series expansion; Lüroth series; Hausdorff dimension,

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Falconer, K. J.: Fractal Geometry, Mathematical Foundations and Application, Wiley, 1990.
Falconer, K. J.: Techniques in Fractal Geometry, Wiley, 1997.
Galambos, J.: The ergodic properties of the denominators in the Oppenheim expansion of real numbers into inŞnite series of rationals, Quart. J. Math. Oxford Sec. Series 21, 177-191 (1970).
Galambos, J.: Further ergodic results on the Oppenheim series, Quart. J. Math. Oxford Sec. Series 25, 135-141 (1974).
Galambos, J.: On the speed of the convergence of the Oppenheim series, Acta Arith. 19, 342(1971).
Galambos, J.: Reprentations of Real Numbers by InŞnite Series, Lecture Notes in Math. , Springer, 1976.
Galambos, J.: Further metric results on series expansions, Publ. Math. Debrecen 52, no. 4, 377-384(1998).
Oppenheim, A.: The representation of real numbers by inŞnite series of rationals, Acta Arith., 21, 391-398 (1972).
Schweiger, F.: Ergodic Theory of Fibred Systems and Metric Number Theory, Oxford, Clarendon Press, 1995.
Vervaat, W.: Success Epochs in Bernoulli Trials, Mathematical Center Tracts 42, Amster- dam, Mathematisch Centrum, 1972.
Wang, B. W., Wu, J.: The Oppenheim series expansions and a problem of Galambos, Publ. Math. Debrecen, to appear. Wu, J.: The Oppenheim series expansions and Hausdorﬀ dimensions, Acta Arith., (2003), 345-355.
Lu-ming SHEN, Yue-hua LIU and Yu-yuan ZHOU Science college of Hunan Agriculture UniversityChangsha, Hunan, 410086, P.R. CHINA