Digital expansion of exponential sequences
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) 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) 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.

@article{JTNB_2002__14_2_477_0,
     author = {Michael Fuchs},
     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},
     zbl = {1072.11006},
     mrnumber = {2040688},
     language = {en},
     url = {https://jtnb.centre-mersenne.org/item/JTNB_2002__14_2_477_0/}
}
TY  - JOUR
AU  - Michael Fuchs
TI  - Digital expansion of exponential sequences
JO  - Journal de théorie des nombres de Bordeaux
PY  - 2002
SP  - 477
EP  - 487
VL  - 14
IS  - 2
PB  - Université Bordeaux I
UR  - https://jtnb.centre-mersenne.org/item/JTNB_2002__14_2_477_0/
LA  - en
ID  - JTNB_2002__14_2_477_0
ER  - 
%0 Journal Article
%A Michael Fuchs
%T Digital expansion of exponential sequences
%J Journal de théorie des nombres de Bordeaux
%D 2002
%P 477-487
%V 14
%N 2
%I Université Bordeaux I
%U https://jtnb.centre-mersenne.org/item/JTNB_2002__14_2_477_0/
%G en
%F JTNB_2002__14_2_477_0
Michael Fuchs. Digital expansion of exponential sequences. Journal de théorie des nombres de Bordeaux, Tome 14 (2002) no. 2, pp. 477-487. https://jtnb.centre-mersenne.org/item/JTNB_2002__14_2_477_0/

[1] A. Baker, Transcendental number theory. Cambridge University Press, Cambridge - New York - Port Chester - Melbourne - Sydney, 1990. | MR | Zbl

[2] A. Baker, G. Wüstholz, Logarithmic forms and group varieties. J. Reine Angew. Math. 442 (1993), 19-62. | MR | Zbl

[3] G. Barat, R.F. Tichy, R. Tijdeman, Digital blocks in linear numeration systems. In Number Theory in Progress (Proceedings of the Number Theory Conference Zakopane 1997, K. Gyôry, H. Iwaniec, and J. Urbanowicz edt.), de Gruyter, Berlin, New York, 1999, 607-633. | MR | Zbl

[4] N.L. Bassily, I. Kàtai, Distribution of the values of q-additive functions on polynomial sequences. Acta Math. Hung. 68 (1995), 353-361. | MR | Zbl

[5] R. Blecksmith, M. Filaseta, C. Nicol, A result on the digits of an. Acta Arith. 64 (1993), 331-339. | MR | Zbl

[6] M. Drmota, R.F. Tichy, Sequences, Discrepancies and Applications. Lecture Notes Math. 1651, Springer, 1997. | MR | Zbl

[7] P. Erdös, P. Turán, On a problem in the theory of uniform distributions I, II. Indagationes Math. 10 (1948), 370-378, 406-413. | MR

[8] P. Kiss, R.F. Tichy, A discrepancy problem with applications to linear recurrences I,II. Proc. Japan Acad. Ser. A Math. Sci. 65 (1989), no. 5, 135-138; no. 6, 191-194. | MR | Zbl

[9] N.M. Korobov, Trigonometric sums with exponential functions and the distribution of signs in repeating decimals. Mat. Zametki 8 (1970), 641-652 = Math. Notes 8 (1970), 831-837. | MR | Zbl

[10] N.M. Korobov, On the distribution of digits in periodic fractions. Matem. Sbornik 89 (1972), 654-670. | MR | Zbl

[11] N.M. Korobov, Exponential sums and their applications. Kluwer Acad. Publ., North-Holland, 1992. | MR | Zbl

[12] H. Niederreiter, On the Distribution of Pseudo-Random Numbers Generated by the Linear Congruential Method II. Math. Comp. 28 (1974), 1117-1132. | MR | Zbl

[13] H. Niederreiter, On the Distribution of Pseudo-Random Numbers Generated by the Linear Congruential Method III. Math. Comp. 30 (1976), 571-597. | MR | Zbl