Digital expansion of exponential sequences

Fuchs, Michael

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

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

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