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Multiplicative properties of the Thue-Morse sequence

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Abstract

The aim of this paper is to give a survey on recent results concerning multiplicative properties of a special sequence of integers, mostly known as the “Thue-Morse sequence”, consisting of integers whose diadic expansion contains an even number of 1. Let us denote by U the associated set

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REFERENCES

  1. S. I. Adian, The Burnside problem and identities in groups, Ergebnisse der Mathematik und ihrer Grenzgebiete 95, Springer-Verlag, 1979.

  2. J.-P. Bertrandias, Suites pseudo-aléatoires et critères d'éequirépartition modulo un, Compositio Math. 16 (1964), 23–28.

    Google Scholar 

  3. J.-P. Bertrandias, Espaces de fonctions bornées et continues en moyenne asymptotique d'ordre p, Bull. Soc. Math. France, Mém. 5 (1966), 1–106.

    Google Scholar 

  4. J. BÉsineau, Indépendance statistique liés à la somme des chiffres, Acta Arith. 20 (1972), 401–416.

    Google Scholar 

  5. R. Bellman and H. N. Shapiro, On a problem in additive number theory, Annals of Math. 49 (1948), 333–340.

    Google Scholar 

  6. J. Coquet, T. Kamae et M. MendÈs-France, Sur la mesure spectrale de certaines suites arithmétiques, Bull. Soc. Math. France 105 (1977), 369–384.

    Google Scholar 

  7. J. Coquet, A summation formula related to binary digits, Invent. Math. 73 (1983), 107–115.

    Google Scholar 

  8. F. M. Dekking, On the distribution of digits in arithmetic sequences, Seminar on number theory, 1982–1983 (Talence, 1982/1983), Exp. No. 32, 12 pp., Univ. Bordeaux I, Talence, 1983.

    Google Scholar 

  9. J. M. Dumont, Discrépance des progressions arithmétiques dans la suite de Morse, C. R. Acad. Sci. Paris, Sér. I 297 (1983), 145–148.

    Google Scholar 

  10. P. D. T. A. Elliott, Probabilistic Number Theory, I, Mean-Value Theorems, Grundlehren Math. Wiss. 239, Springer, 1979.

  11. P. ErdŐs, C. Pomerance, A. SÁrkÖzy and C. L. Stewart, On elements of sumsets with many prime factors, J. Number Theory 44 (1993), 93–104.

    Google Scholar 

  12. C. G. Esseen, Fourier analysis of distribution functions. A mathematical study of Laplace-Gaussian law, Acta Math. 77 (1945), 1–125.

    Google Scholar 

  13. M. N. J. Fine, The distribution of the sum of digits (mod p), Bull Amer. Math. Soc. 71 (1965), 2651–2652.

    Google Scholar 

  14. E. Fouvry et C. Mauduit, Somme des chiffres et nombres presque premiers, Math. Ann. 305 (1996), 571–599.

    Google Scholar 

  15. E. Fouvry et C. Mauduit, Méthodes de crible et fonction somme des chiffres, Acta Arith. 77 (1996) 339–351.

    Google Scholar 

  16. A. O. Gelfond, Sur les nombres qui ont des propriétés additives et multiplicatives données, Acta Arith. 13 (1968), 259–265.

    Google Scholar 

  17. H. Halberstam and H. E. Richert, Sieve methods, Academic Press, New York, 1974.

    Google Scholar 

  18. G. Harman and J. Rivat, Primes of the form [p c] and related questions, Glasgow Mathematical Journal 37 (1995), 131–141.

    Google Scholar 

  19. L. HervÉ, Étude d'opérateurs quasi-compacts positifs. Applications aux opérateurs de transfert, Ann. Inst. Henri Poincaré 30 (1994), 437–466.

    Google Scholar 

  20. L. HervÉ, Construction et régularité des fonctions d'échelles, SIAM J. Anal. Math. 26 (1995), 1367–1385.

    Google Scholar 

  21. H. Iwaniec and J. Pomykala, Sums and differences of quartic norm, Mathematika 40 (1993), 233–245.

    Google Scholar 

  22. T. Kamae, Sum of digits to different bases and mutual singularity of their spectral measures, Osaka J. Math. 15 (1978), 569–574.

    Google Scholar 

  23. T. Kamae, Mutual singularity of spectra of dynamical systems given by sums of digits to different bases, Systèmes dynamiques I, Varsovie, Astérisque 49 (1978), 109–116.

    Google Scholar 

  24. M. Keane, Generalized Morse sequences, Zeitschr. Wahr. 10 (1968), 335–353.

    Google Scholar 

  25. D. H. Kim, On joint distribution of q-additive functions in residue classes, Journal of Number Theory 74 (1999), 307–336.

    Google Scholar 

  26. M. Lothaire, Combinatorics on words, Encyclopedia of Mathematics and its Applications, 17, Addison-Wesley, 1983.

  27. K. Mahler, The spectrum of an array and its application to the study of the translation properties of a simple class of arithmetical functions II, On the translation properties of a simple class of arithmetical functions, J. Math. Massachusetts 6 (1927), 158–163.

    Google Scholar 

  28. M. MendÈs-France, Nombres normaux, Application aux fonctions pseudo-aléatoires, J. Anal. Math. 20 (1967), 1–56.

    Google Scholar 

  29. C. Mauduit and B. MossÉ, Suites de G q-orbite finie, Acta Arith. 57 (1991), 69–82.

    Google Scholar 

  30. M. Morse, Recurrent geodesics on a surface of negative curvature, Trans. Amer. Math. Soc. 22 (1921), 84–100.

    Google Scholar 

  31. C. Mauduit and J. Rivat, Répartition des fonctions q-multiplicatives dans la suite [n c], Acta Arith. 71 (1995), 171–179.

    Google Scholar 

  32. C. Mauduit and A. SÁrkÖzy, On the arithmetic structure of sets characterized by sum of digits properties, J. Number Theory 61 (1996), 25–38.

    Google Scholar 

  33. C. Mauduit and A. SÁrkÖzy, On the arithmetic structure of integers whose sum of digits is fixed, Acta Arith. 81 (1997), 145–173.

    Google Scholar 

  34. D. J. Newman, On the number of binary digits in a multiple of three, Proc. Amer. Math. Soc. 21 (1969), 719–721.

    Google Scholar 

  35. D. J. Newman and M. Slater, Binary digit distribution over naturally defined sequences, Trans. Amer. Math. Soc. 213 (1975)

  36. E. Prouhet, Mémoire sur quelques relations entre les puissances des nombres, C. R. Acad. Sc. Paris 33 (1851), 31.

    Google Scholar 

  37. I. I. Piatetski-Shapiro, On the distribution of prime numbers in sequences of the form [f(m)], Mat. Sb. 33 (1953), 559–566.

    Google Scholar 

  38. M. Queffelec, Sur la singularité des produits de Riesz et des mesures spectrales associées à la somme de chiffres, Israel J. Math. 34 (1979), 337–342.

    Google Scholar 

  39. M. Queffelec, Substitution dynamical systems-Spectral analysis, Lecture Notes in Mathematics 1294, Springer Verlag, Berlin, 1987.

    Google Scholar 

  40. G. Rauzy, Propriétés statistiques des suites arithmétiques, collection SUP, Presses Universitaires de France, Paris, 1976.

    Google Scholar 

  41. J. Rivat and P. Sargos, Nombres premiers de la forme [n c], preprint.

  42. G. Tenenbaum, Sur la non-dérivabilité de fonctions périodiques associées à certaines formules sommatoires, The mathematics of Paul Erdős, Algorithms and Combinatorics 13 Springer-Verlag, 1997, 117–128.

  43. A. Thue, Über unendliche Zeichenreihen (1906), Selected mathematical papers of Axel Thue, Universitetsforlaget, 1977.

  44. A. THUE, Über die gegenseitige Lage gleicher Teile gewisser Zeichenreihen (1912), Selected mathematical papers of Axel Thue, Universitetsforlaget, 1977.

  45. N. Wiener, The spectrum of an array and its application to the study of the translation properties of a simple class of arithmetical function I. The spectrum of an array, J. Math. Massachusetts 6 (1927), 145–157.

    Google Scholar 

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  1. Institut de Mathématiques de Luminy, CNRS-UPR 9016, 163, Avenue de Luminy, Case 930, 13288, Marseille Cedex 9, France

    Christian Maudutt

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Maudutt, C. Multiplicative properties of the Thue-Morse sequence. Periodica Mathematica Hungarica 43, 137–153 (2002). https://doi.org/10.1023/A:1015241900975

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