On the range of fractional parts {ξ(p/q)ⁿ}

Leopold Flatto; Jeffrey C. Lagarias; Andrew D. Pollington

Acta Arithmetica (1995)

  • Volume: 70, Issue: 2, page 125-147
  • ISSN: 0065-1036

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Leopold Flatto, Jeffrey C. Lagarias, and Andrew D. Pollington. "On the range of fractional parts {ξ(p/q)ⁿ}." Acta Arithmetica 70.2 (1995): 125-147. <http://eudml.org/doc/206742>.

@article{LeopoldFlatto1995,
author = {Leopold Flatto, Jeffrey C. Lagarias, Andrew D. Pollington},
journal = {Acta Arithmetica},
keywords = {distribution modulo one; distribution of the fractional parts; powers of a non-integer rational number; symbolic dynamics},
language = {eng},
number = {2},
pages = {125-147},
title = {On the range of fractional parts \{ξ(p/q)ⁿ\}},
url = {http://eudml.org/doc/206742},
volume = {70},
year = {1995},
}

TY - JOUR
AU - Leopold Flatto
AU - Jeffrey C. Lagarias
AU - Andrew D. Pollington
TI - On the range of fractional parts {ξ(p/q)ⁿ}
JO - Acta Arithmetica
PY - 1995
VL - 70
IS - 2
SP - 125
EP - 147
LA - eng
KW - distribution modulo one; distribution of the fractional parts; powers of a non-integer rational number; symbolic dynamics
UR - http://eudml.org/doc/206742
ER -

References

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  1. M. J. Bertin, A. Decomps-Guilloux, M. Grandet-Hugot, M. Pathiaux-Delefosse and J. P. Schreiber (1992), Pisot and Salem Numbers, Birkhäuser, Boston. Zbl0772.11041
  2. D. Boyd (1969), Transcendental numbers with badly distributed powers, Proc. Amer. Math. Soc. 23, 424-427. Zbl0186.08704
  3. G. Choquet (1981), θ-fermés et dimension de Hausdorff. Conjectures de travail. Arithmétique des θ-cycles (θ = 3/2), C. R. Acad. Sci. Paris Sér. I 292, 339-344. Zbl0465.10042
  4. L. Flatto (1992), Z-numbers and β-transformations, in: Symbolic Dynamics and Its Applications, P. Walters (ed.), Contemp. Math. 135, Amer. Math. Soc., Providence, R.I., 181-202. 
  5. L. Flatto and J. C. Lagarias (1994), The lap counting function for linear mod one transformations, I. Explicit formulas and renormalizability, Ergodic Theory Dynamical Systems, to appear. Zbl0865.58016
  6. L. Flatto, J. C. Lagarias and B. Poonen (1994), The zeta function of the beta transformation, Ergodic Theory Dynamical Systems 14, 237-266. Zbl0843.58106
  7. K. Mahler (1968), An unsolved problem on the powers of 3/2, J. Austral. Math. Soc. 8, 313-321. Zbl0155.09501
  8. W. Parry (1960), On the β-expansion of real numbers, Acta Math. Acad. Sci. Hungar. 11, 401-416. Zbl0099.28103
  9. C. Pisot (1938), La répartition modulo 1 et les nombres algébriques, Ann. Scuola Norm. Sup. Pisa (2) 7, 204-248. Zbl64.0994.01
  10. C. Pisot (1946), Répartition (mod 1) des puissances successives des nombres réels, Comment. Math. Helv. 19, 153-160. Zbl0063.06259
  11. A. Pollington (1978), Intervals Constructions in the Theory of Numbers, Ph.D. thesis, University of London. 
  12. A. D. Pollington (1979), On the density of sequences {nₖξ}, Illinois J. Math. 23, 511-515. Zbl0413.10042
  13. A. D. Pollington (1981), Progressions arithmétiques généralisées et le problème des (3/2)ⁿ, C. R. Acad. Sci. Paris Sér. I 292, 383-384. Zbl0466.10038
  14. A. Rényi (1957), Representations for real numbers and their ergodic properties, Acta Math. Acad. Sci. Hungar. 8, 472-493. Zbl0079.08901
  15. R. Tijdeman (1972), Note on Mahler's 3/2-Problem, K. Norske Vidensk. Selsk. Skr. 16, 1-4. 
  16. T. Vijayaraghavan (1940), On the fractional parts of the powers of a number, I, J. London Math. Soc. 15, 159-160. 
  17. T. Vijayaraghavan (1941), On the fractional parts of the powers of a number, II, Proc. Cambridge Philos. Soc. 37, 349-357. Zbl67.0988.02
  18. T. Vijayaraghavan (1942), On the fractional parts of the powers of a number, III, J. London Math. Soc. 17, 137-138. Zbl0060.12204
  19. T. Vijayaraghavan (1948), On the fractional parts of powers of a number, IV, J. Indian Math. Soc. (N.S.) 12, 33-39. Zbl0031.11502

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