Abstract
In a previous paper, [7], the authors together with Gavin Brown gave a complete description of the values ofθ, r ands for which numbers normal in baseθ r are normal in baseθ s. Hereθ is some real number greater than 1 andx is normal in baseθ if {θ n x} is uniformly distributed modulo 1. The aim of this paper is to complete this circle of ideas by describing thoseφ andψ for which normality in baseφ implies normality in baseψ. We show, in fact, that this can only happen if both are integer powers of some baseθ and are thus subject to the constraints imposed by the results of [7]. This paper then completes the answer to the problem raised by Mendès France in [12] of determining thoseφ andψ for which normality in one implies normality in the other.
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References
Anne Bertrand,Répartition modulo un des suites exponentielles et de dynamiques symboliques, Ph.D. thesis, L’Universite de Bordeaux I, 1986.
Anne Bertrand,Dévelopements en base de Pisot et répartition modulo un de la suite (xθ n) n≥0 ,langages codés et θ-shift, Bulletin de la Société Mathématique de France114 (1986), 271–323.
G. Brown and W. Moran,Schmidt’s Conjecture on normality with respect to commuting matrices, Inventiones Mathematicae111 (1993), 449–463.
G. Brown, W. Moran and C.E.M. Pearce,Riesz products and normal numbers, Journal of the London Mathematical Society32 (1985), 12–18.
G. Brown, W. Moran and C.E.M. Pearce,A decomposition theorem for numbers in which the summands have prescribed normality properties, Journal of Number Theory24 (1986), 259–271.
G. Brown, W. Moran and C.E.M. Pearce,Riesz products, Hausdorff dimension and normal numbers, Mathematical Proceedings of the Cambridge Philosophical Society101 (1987), 529–540.
G. Brown, W. Moran and A. D. Pollington,Normality to powers of a base, submitted.
J.W.S. Cassels,On a problem of Steinhaus about normal numbers, Colloquium Mathematicum7 (1959), 95–101.
C.C. Graham and O.C. McGehee,Essays in Commutative Harmonic Analysis, Grundelehren der mathematischen Wissenschaften Vol. 238, Springer-Verlag, New York, 1979.
J. Feldman and M. Smorodinsky,Normal numbers from independent processes, Ergodic Theory and Dynamical Systems12 (1992), 707–712.
William J. LeVeque,Fundamentals of Number Theory, Addison-Wesley, 1977.
M. Mendès France,Les Ensembles de Bésineau, Séminaire Delange-Pisot-Poitou. Théorie des nombres, Vol. 1 Fasc. 7, Secrétariat Mathématique, Paris, 1975.
W. Moran and A.D. Pollington,Metrical results on normality to distinct bases, Journal of Number Theory54 (1995), 180–189.
W. Moran and A.D. Pollington,Normality to non-integer bases, inAnalytic Number Theory and Related Topics, World Scientific, Singapore, 1993, pp. 109–117.
W. Moran and A.D. Pollington,A problem in multi-index normality, Fourier Analysis and Its Applications: Kahane Special Issue (1995), 455–466.
G. Rauzy,Propriétés statistiques de suites arithmétiques, Presses Universitaires de France, 1976.
W. M. Schmidt,On normal numbers, Pacific Journal of Mathematics10 (1960), 661–672.
W. M. Schmidt,Über die Normalität von Zahlen zu verschiedenen Basen, Acta Arithmetica7 (1962), 299–301.
H. Steinhaus,The New Scottish Book, Problem 144, Wroclaw, 1958, p. 14.
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Moran, W., Pollington, A.D. The discrimination theorem for normality to non-integer bases. Isr. J. Math. 100, 339–347 (1997). https://doi.org/10.1007/BF02773647
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DOI: https://doi.org/10.1007/BF02773647