Hostname: page-component-cd9895bd7-hc48f Total loading time: 0 Render date: 2024-12-27T10:46:27.070Z Has data issue: false hasContentIssue false
  • English
  • Français

ABSOLUTELY ABNORMAL AND CONTINUED FRACTION NORMAL NUMBERS

Part of: Probabilistic theory: distribution modulo $1$; metric theory of algorithms Elementary number theory Diophantine approximation, transcendental number theory

Published online by Cambridge University Press:  16 March 2016

JOSEPH VANDEHEY
JOSEPH VANDEHEY*
Affiliation:
Department of Mathematics, University of Georgia, Athens, GA 30602, USA email vandehey@uga.edu
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In this short note, we give a proof, conditional on the generalised Riemann hypothesis, that there exist numbers $x$ which are normal with respect to the continued fraction expansion but not to any base-$b$ expansion. This partially answers a question of Bugeaud.

Keywords

continued fractionsnormal numbersArtin’s conjecture

MSC classification

Primary: 11K16: Normal numbers, radix expansions, Pisot numbers, Salem numbers, good lattice points, etc.
Secondary: 11J70: Continued fractions and generalizations 11A07: Congruences; primitive roots; residue systems
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 94 , Issue 2 , October 2016 , pp. 217 - 223
Copyright
© 2016 Australian Mathematical Publishing Association Inc. 

References

Adler, R., Keane, M. and Smorodinsky, M., ‘A construction of a normal number for the continued fraction transformation’, J. Number Theory 13(1) (1981), 95–105.Google Scholar
Bailey, D. H. and Borwein, J. M., ‘Nonnormality of Stoneham constants’, Ramanujan J. 29(1–3) (2012), 409–422.Google Scholar
Bugeaud, Y., Distribution Modulo One and Diophantine Approximation, Cambridge Tracts in Mathematics, 193 (Cambridge University Press, Cambridge, 2012).CrossRefGoogle Scholar
Cassels, J. W. S., ‘On a problem of Steinhaus about normal numbers’, Colloq. Math. 7 (1959), 95–101.Google Scholar
Champernowne, D. G., ‘The construction of decimals normal in the scale of ten’, J. Lond. Math. Soc. (2) 8 (1933), 254–260.CrossRefGoogle Scholar
Dajani, K. and Kraaikamp, C., Ergodic Theory of Numbers, Carus Mathematical Monographs, 29 (Mathematical Association of America, Washington, DC, 2002).Google Scholar
Kraaikamp, C. and Nakada, H., ‘On normal numbers for continued fractions’, Ergod. Th. & Dynam. Sys. 20(5) (2000), 1405–1421.Google Scholar
Lenstra, H. W. Jr, ‘On Artin’s conjecture and Euclid’s algorithm in global fields’, Invent. Math. 42 (1977), 201–224.Google Scholar
Mance, B., ‘Number theoretic applications of a class of Cantor series fractal functions. I’, Acta Math. Hungar. 144(2) (2014), 449–493.Google Scholar
Martin, G., ‘Absolutely abnormal numbers’, Amer. Math. Monthly 108(8) (2001), 746–754.Google Scholar
Maxfield, J. E., ‘Normal k-tuples’, Pacific J. Math. 3 (1953), 189–196.Google Scholar
Moree, P., ‘On primes in arithmetic progression having a prescribed primitive root’, J. Number Theory 78(1) (1999), 85–98.CrossRefGoogle Scholar
Moree, P., ‘On primes in arithmetic progression having a prescribed primitive root. II’, Funct. Approx. Comment. Math. 39(1) (2008), 133–144.Google Scholar
Schmidt, W. M., ‘Über die Normalität von Zahlen zu verschiedenen Basen’, Acta Arith. 7 (1961–1962), 299–309.CrossRefGoogle Scholar
Steinhaus, H., ‘Problem 144’, in: The New Scottish Book: Wrocław 1946–1958, http://www.wmi.uni.wroc.pl/New_Scottish_Book.Google Scholar
Vandehey, J., ‘On the joint normality of certain digit expansions’, Preprint, 2014, arXiv:1408.0435.Google Scholar