Digital expansion of exponential sequences

Fuchs, Michael

Fuchs, Michael

Journal de théorie des nombres de Bordeaux, Tome 14 (2002) no. 2, pp. 477-487.

Résumé
Abstract

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 ${(log N)}^{\frac{3}{2} - ϵ}$ , avec $ϵ > 0$ , premiers termes de la suite $a^{n}$ .

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 $c log N$ digits is asymptotically equal to the expected value. Under stronger assumptions we get a similar result for the first ${(log N)}^{\frac{3}{2} - ϵ}$ 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.

MR Zbl

@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},
     pages = {477--487},
     publisher = {Universit\'e Bordeaux I},
     volume = {14},
     number = {2},
     year = {2002},
     mrnumber = {2040688},
     zbl = {1072.11006},
     language = {en},
     url = {https://www.numdam.org/item/JTNB_2002__14_2_477_0/}
}

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Fuchs, Michael. Digital expansion of exponential sequences. Journal de théorie des nombres de Bordeaux, Tome 14 (2002) no. 2, pp. 477-487. https://www.numdam.org/item/JTNB_2002__14_2_477_0/

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