Hostname: page-component-6bf8c574d5-685pp Total loading time: 0 Render date: 2025-03-07T09:10:15.648Z Has data issue: false hasContentIssue false
  • English
  • Français

On the Powers of Some Transcendental Numbers

Published online by Cambridge University Press:  17 April 2009

Artūras Dubickas
Artūras Dubickas
Affiliation:
Department of Mathematics and Informatics, Vilnius University, Naugarduko 24, Vilnius LT-03225, Lithuania, e-mail: [email protected]
Rights & Permissions [Opens in a new window]

Extract

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.

We construct a transcendental number α whose powers αn!, n = 1, 2, 3,…, modulo 1 are everywhere dense in the interval [0, 1]. Similarly, for any sequence of positive numbers δ = (δn)∞n=1, we find a transcendental number α = α(δ) such that the inequality {αn} < δn holds for infinitely many n ∈ N, no matter how fast the sequence δ converges to zero. Finally, for any sequence of real numbers (rn)∞n=1 and any sequence of positive numbers (δn)∞n=1, we construct an increasing sequence of positive integers (qn)∞n=1 and a number α > 1 such that ‖αqn – τn‖ < δn for each n ≥ 1.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 76 , Issue 3 , December 2007 , pp. 433 - 440
Copyright
Copyright © Australian Mathematical Society 2007

References

[1]Adhikari, S.D. and Rath, P., ‘A problem of the fractional parts of the powers of 3/2 and related questions’, in Proceedings of a Number Theory Conference held in Chandigarh, 2005 (to appear).Google Scholar
[2]Alkauskas, G. and Dubickas, A., ‘Prime and composite numbers as integer parts of powers’, Acta Math. Hungar. 105 (2004), 249–256.CrossRefGoogle Scholar
[3]Bugeaud, Y. and Dubickas, A., ‘Fractional parts of powers and Sturmian words’, C. R. Math. Acad. Sci. Paris 341 (2005), 69–74.CrossRefGoogle Scholar
[4]Corvaja, P. and Zannier, U., ‘On the rational approximations to the powers of an algebraic number: Solution of two problems of Mahler and Mendès France’, Acta Math. 193 (2004), 175–191.CrossRefGoogle Scholar
[5]Dubickas, A., ‘A note on powers of Pisot numbers’, Publ. Math. Debrecen 56 (2000), 141–144.CrossRefGoogle Scholar
[6]Dubickas, A., ‘On the limit points of the fractional parts of powers of Pisot numbers’, Arch. Math. 42 (2006), 151–158.Google Scholar
[7]Koksma, J.F., ‘Ein mengentheoretischer Satz über Gleichverteilung modulo eins’, Compositio Math. 2 (1935), 250–258.Google Scholar
[8]Lerma, M.A., ‘Construction of a number greater than one whose powers are uniformy distributed modulo one’, (http://www.math.northwestern.edu/~mlerma/papers/constr_ud_mod1.pdf) (1996).Google Scholar
[9]Luca, F., ‘On a question of G. Kubaz’, Arch. Math. (Basel) 74 (2000), 269–275.CrossRefGoogle Scholar
[10]Mahler, K., ‘On the fractional parts of the powers of a rational number, II’, Mathematika 4 (1957), 122–124.CrossRefGoogle Scholar
[11]Mahler, K., ‘An unsolved problem on the powers of 3/2’, J. Austral. Math. Soc. 8 (1968), 313–321.CrossRefGoogle Scholar
[12]Pisot, Ch., ‘La répartition modulo 1 et les nombres algébriques’, Ann. Sc. Norm. Sup. Pisa 7 (1938), 205–248.Google Scholar
[13]Pisot, Ch. and Salem, R., ‘Distribution modulo 1 of the powers of real numbers larger than 1’, Compositio Math. 16 (1964), 164–168.Google Scholar
[14]Salem, R., Algebraic numbers and Fourier analysis (D. C. Heath and Co., Boston, MA, 1963).Google Scholar
[15]Vijayaraghavan, T., ‘On the fractional parts of the powers of a number. I’, J. London Math. Soc. 15 (1940), 159–160.CrossRefGoogle Scholar
[16]Weyl, H., ‘Über die Gleichverteilung von Zahlen modulo Eins’, Math. Ann. 77 (1916), 313–352.CrossRefGoogle Scholar
[17]Zaimi, T., ‘On integer and fractional parts powers of Salem numbers’, Arch. Math. (Basel) 87 (2006), 124–128.CrossRefGoogle Scholar