Digital expansion of exponential sequences
Journal de théorie des nombres de Bordeaux, Volume 14 (2002) no. 2, p. 477-487

We consider the q-ary digital expansion of the first N terms of an exponential sequence a n . Using a result due to Kiss and Tichy [8], we prove that the average number of occurrences of an arbitrary digital block in the last clogN digits is asymptotically equal to the expected value. Under stronger assumptions we get a similar result for the first (logN) 3 2-ϵ digits, where ϵ is a positive constant. In both methods, we use estimations of exponential sums and the concept of discrepancy of real sequences modulo 1 plays an important role.

On s’intéresse au développement en base q des N premiers termes de la suite exponentielle a n . En utilisant un résultat dû à Kiss et Tichy, nous montrons que le nombre moyen d’occurrences d’un bloc de chiffres donné est égal asymptotiquement à sa valeur supposée. Sous une hypothèse plus forte nous montrons un résultat similaire en ne considérant seulement les (logN) 3 2-ϵ , avec ϵ>0, premiers termes de la suite a n .

@article{JTNB_2002__14_2_477_0,
     author = {Fuchs, Michael},
     title = {Digital expansion of exponential sequences},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     publisher = {Universit\'e Bordeaux I},
     volume = {14},
     number = {2},
     year = {2002},
     pages = {477-487},
     zbl = {1072.11006},
     mrnumber = {2040688},
     language = {en},
     url = {http://www.numdam.org/item/JTNB_2002__14_2_477_0}
}
Fuchs, Michael. Digital expansion of exponential sequences. Journal de théorie des nombres de Bordeaux, Volume 14 (2002) no. 2, pp. 477-487. http://www.numdam.org/item/JTNB_2002__14_2_477_0/

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