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Title

Odometers and systems of numeration

Authors 1, 2, 1

Affiliations

  1. Institut für Mathematik, TU Graz, Steyrergasse 30, 8010 Graz, Austria
  2. DSA, CMI, 39, Rue Joliot Curie, 13453 Marseille, France

Bibliography

  1. [AM] J.-P. Allouche and M. Mendès France, On an extremal property of the Rudin-Shapiro sequence, Mathematika 32 (1985), 33-38.
  2. [Ba1] J. Bass, Fonctions pseudo-aléatoires et fonctions de Wiener, C. R. Acad. Sci. Paris 247 (1958), 1163-1165.
  3. [Ba2] J. Bass, Suites uniformément denses, moyennes trigonométriques, fonctions pseudo-aléatoires, Bull. Soc. Math. France 87 (1959), 1-64.
  4. [Be] A. Bertrand, Développements en base de Pisot et répartition modulo 1, C. R. Acad. Sci. Paris 285 (1977), 419-421.
  5. [Ber] J.-P. Bertrandias, Colloque sur la répartition asymptotique mod. 1, Breukelen, 1962.
  6. [Bl] F. Blanchard, β-expansions and symbolic dynamics, Theoret. Comput. Sci. 65 (1989), 131-141.
  7. [Bo] D. Boyd, Salem numbers of degree four have periodic expansions, in: Number Theory, J.-M. De Koninck and C. Levesque (eds.), W. de Gruyter, 1989, 57-64.
  8. [Br] A. Brauer, On algebraic equations with all but one root in the interior of the unit circle, Math. Nachr. 4 (1951), 250-257.
  9. [Co] J. Coquet, Représentations lacunaires des entières naturelles I,II, Arch. Math. (Basel) 38 (1982), 184-188; 41 (1983), 238-242.
  10. [Du] J.-M. Dumont, Formules sommatoires et systèmes de numération liés aux substitutions, Séminaire de Théorie des Nombres, Bordeaux (1987/88), 12 pp.
  11. [DT1] J.-M. Dumont et A. Thomas, Systèmes de numération et fonctions fractales relatifs aux substitutions, Theoret. Comput. Sci. 65 (1989), 153-169.
  12. [DT2] J.-M. Dumont and A. Thomas, Digital sum problems and substitution on a finite alphabet, preprint.
  13. [Dup] Y. Dupain, Intervalles à restes majorés pour la suite nα, Acta Math. Acad. Sci. Hungar. 29 (1977), 289-303.
  14. P. Flajolet, P. J. Grabner, P. Kirschenhofer, H. Prodinger and R. F. Tichy, Mellin-transform and asymptotics: digital sums, Theoret. Comput. Sci., to appear.
  15. [Fr] A. S. Fraenkel, Systems of numeration, Amer. Math. Monthly 92 (1985), 105-114.
  16. [F1] C. Frougny, Representation of numbers and finite automata, Math. Systems Theory 25 (1992), 37-60.
  17. [F2] C. Frougny, How to write integers in non-integer base, in: LATIN 92, S ao Paulo, Lecture Notes in Comput. Sci. 583, Springer, 1992, 154-164.
  18. [F3] C. Frougny, Fibonacci representations and finite automata, IEEE Trans. Inform. Theory 37 (1991), 393-399.
  19. [FS] C. Frougny and B. Solomyak, Finite beta-expansions, Ergodic Theory Dynamical Systems 12 (1992), 713-723.
  20. P. J. Grabner, P. Kirschenhofer, H. Prodinger and R. F. Tichy, On the moments of the sum-of-digits function, in: Applications of Fibonacci Numbers, G. Bergum, A. F. Horadam and A. N. Philippou (eds.), Vol. 5, Kluwer, 1993, 263-271.
  21. [GT1] P. J. Grabner and R. F. Tichy, Contributions to digit expansions with respect to linear recurrences, J. Number Theory 36 (1990), 160-169.
  22. [GT2] P. J. Grabner and R. F. Tichy, α-expansions, linear recurrences and the sum-of-digits function, Manuscripta Math. 70 (1991), 311-324.
  23. [HR] E. Hewitt and K. A. Ross, Abstract Harmonic Analysis, Springer, Berlin, 1970.
  24. [HW] G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, Oxford University Press, 1954.
  25. [KT] P. Kirschenhofer and R. F. Tichy, On the distribution of digits in Cantor-representations of integers, J. Number Theory 18 (1984), 121-134.
  26. [Kn] D. E. Knuth, Fibonacci multiplication, Appl. Math. Lett. 1 (1988), 57-60.
  27. [Ko] N. Kopecek, Zifferndarstellungen natürlicher Zahlen und Diskrepanzabschätzungen, 1991, Thesis, TU Wien.
  28. N. Kopecek, G. Larcher, R. F. Tichy and G. Turnwald, On the discrepancy of sequences associated with the sum-of-digits function, Ann. Inst. Fourier (Grenoble) 37 (1987), 1-17.
  29. [KN] L. Kuipers and H. Niederreiter, Uniform Distribution of Sequences, J. Wiley, New York, 1974.
  30. [La] Y. Lacroix, Remarks on the Delange-Coquet formula, Anz. Österreich. Akad. Wiss., to appear.
  31. [Li1] P. Liardet, Propriétés Harmoniques de la Numération suivant Jean Coquet, Publ. Math. d'Orsay N° 88-02, Colloque de Théorie des Nombres 'Jean Coquet', 1-35.
  32. [Li2] P. Liardet, Regularities of distribution, Compositio Math. 61 (1987), 267-293.
  33. [Li3] P. Liardet, Dynamical properties of the Ostrowski α-expansion, preprint.
  34. [Ma] K. Mahler, On the translation properties of a simple class of arithmetical functions, J. Math. Phys. 6 (1927), 158-163.
  35. [Me1] M. Mendès France, Nombres normaux, applications aux fonctions pseudo-aléatoires, J. Analyse Math. 20 (1967), 1-56.
  36. [Me2] M. Mendès France, Nombres normaux et fonctions pseudo-aléatoires, Ann. Inst. Fourier (Grenoble) 13 (1963), 91-104.
  37. [Me3] M. Mendès France, Les suites à spectre vide et la répartition modulo 1, J. Number Theory 5 (1973), 1-15.
  38. [Os] A. Ostrowski, Bemerkungen zur Theorie der Diophantischen Approximationen I, II, Abh. Math. Sem. Hamburg 1 (1922), 77-98, 250-251.
  39. [Pa] W. Parry, On the β-expansions of real numbers, Acta Math. Acad. Sci. Hungar. 11 (1960), 401-416.
  40. [PT] A. Pethő and R. F. Tichy, On digit expansions with respect to linear recurrences, J. Number Theory 33 (1989), 243-256.
  41. [Pi] C. Pisot, La répartition modulo 1 et les nombres algébriques, Ann. Scuola Norm. Sup. Pisa 2 (1938), 205-248.
  42. [Qu] M. Queffélec, Substitution Dynamical Systems - Spectral Analysis, Lecture Notes in Math. 1294, Springer, 1987.
  43. [Ra] G. Rauzy, Nombres algébriques et substitutions, Bull. Soc. Math. France 110 (1982), 147-178.
  44. [Ré] A. Rényi, Representation for real numbers and their ergodic properties, Acta Math. Acad. Sci. Hungar. 8 (1957), 477-493.
  45. [Sch] K. Schmidt, On periodic expansion of Pisot numbers and Salem numbers, Bull. London Math. Soc. 12 (1980), 269-278.
  46. [Sg] J. Schoissengeier, On the discrepancy of nα, Acta Arith. 44 (1984), 241-279.
  47. [So1] B. Solomyak, On the spectral theory of adic transformations, Adv. Soviet Math. 9 (1992).
  48. [So2] B. Solomyak, Substitutions, adic transformations and beta-expansions, in: Contemporary Mathematics (Proc. of the Conference on Symbolic Dynamics, Yale, 1991), to appear.
  49. [So3] B. Solomyak, On a dynamical system with discrete spectrum, Russian Math. Surveys 41 (1986), 219-220.
  50. [Sós] V. T. Sós, On the distribution of the sequence nα, Ann. Univ. Sci. Budapest Eötvös Sect. Math. 1 (1958), 127-134.
  51. [TT1] R. F. Tichy and G. Turnwald, On the discrepancy of some special sequences, J. Number Theory 26 (1987), 68-78.
  52. [TT2] R. F. Tichy and G. Turnwald, Gleichmäß ige Diskrepanzabschätzungen für Ziffernsummen, Österr. Akad. Wiss. SB II (1986), 17-21.
  53. [Ve] A. M. Vershik, Uniform algebraic approximation of shift and multiplication operators, Soviet Math. Dokl. 24 (1981), 97-100.
  54. [Wa] P. Walters, Ergodic Theory, Springer, Berlin, 1982.
  55. [Wi] N. Wiener, The spectrum of an array and its applications to the study of the translation properties of a simple class of arithmetical functions, J. Math. Phys. 6 (1927), 145-157.
Acta Arithmetica
1995/70/2
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Pages:
103-123
Main language of publication
English
Received
1993-07-02
Accepted
1993-12-07
Published
1995
Exact and natural sciences