# Article

MSC: 11A63, 11K16
Full entry | Fulltext not available (moving wall 24 months)
Keywords:
Cantor series; normal number
Summary:
Let $Q=(q_n)_{n=1}^\infty$ be a sequence of bases with $q_i\ge 2$. In the case when the $q_i$ are slowly growing and satisfy some additional weak conditions, we provide a construction of a number whose $Q$-Cantor series expansion is both $Q$-normal and $Q$-distribution normal. Moreover, this construction will result in a computable number provided we have some additional conditions on the computability of $Q$, and from this construction we can provide computable constructions of numbers with atypical normality properties.
References:
[1] Airey, D., Mance, B.: On the Hausdorff dimension of some sets of numbers defined through the digits of their $Q$-Cantor series expansions. J. Fractal Geom. 3 (2016), 163-186. DOI 10.4171/JFG/33 | MR 3501345
[2] Airey, D., Mance, B.: Normal equivalencies for eventually periodic basic sequences. Indag. Math., New Ser. 26 (2015), 476-484. DOI 10.1016/j.indag.2015.02.002 | MR 3341809 | Zbl 1326.11036
[3] Airey, D., Mance, B., Vandehey, J.: Normality preserving operations for Cantor series expansions and associated fractals II. New York J. Math. (electronic only) 21 (2015), 1311-1326. MR 3441645
[4] Altomare, C., Mance, B.: Cantor series constructions contrasting two notions of normality. Monatsh. Math. 164 (2011), 1-22. DOI 10.1007/s00605-010-0213-0 | MR 2827169 | Zbl 1276.11128
[5] Becher, V., Figueira, S., Picchi, R.: Turing's unpublished algorithm for normal numbers. Theor. Comput. Sci. 377 (2007), 126-138. DOI 10.1016/j.tcs.2007.02.022 | MR 2323391 | Zbl 1117.03051
[6] Cantor, G.: Über die einfachen Zahlensysteme. Zeitschrift für Mathematik und Physik 14 (1869), 121-128 German.
[7] Champernowne, D. G.: The construction of decimals normal in the scale of ten. J. Lond. Math. Soc. 8 (1933), 254-260. DOI 10.1112/jlms/s1-8.4.254 | MR 1573965 | Zbl 0007.33701
[8] Erdős, P., Rényi, A.: On Cantor's series with convergent $\sum 1/q_n$. Ann. Univ. Sci. Budap. Rolando Eötvös, Sect. Math. 2 (1959), 93-109. MR 0126414 | Zbl 0095.26501
[9] Erdős, P., Rényi, A.: Some further statistical properties of the digits in Cantor's series. Acta Math. Acad. Sci. Hung. 10 (1959), 21-29. DOI 10.1007/BF02063287 | MR 0107631 | Zbl 0088.25804
[10] Galambos, J.: Representations of Real Numbers by Infinite Series. Lecture Notes in Mathematics 502 Springer, Berlin (1976). MR 0568141 | Zbl 0322.10002
[11] Kuipers, L., Niederreiter, H.: Uniform Distribution of Sequences. Pure and Applied Mathematics John Wiley & Sons, New York (1974). MR 0419394 | Zbl 0281.10001
[12] Mance, B.: Number theoretic applications of a class of Cantor series fractal functions I. Acta Math. Hung. 144 (2014), 449-493. DOI 10.1007/s10474-014-0456-7 | MR 3274409 | Zbl 1320.11069
[13] Mance, B.: Construction of normal numbers with respect to the $Q$-Cantor series expansion for certain $Q$. Acta Arith. 148 (2011), 135-152. MR 2786161 | Zbl 1239.11082
[14] Rényi, A.: Probabilistic methods in number theory. Proc. Int. Congr. Math. (1958), 529-539. MR 0118707
[15] Rényi, A.: On the distribution of the digits in Cantor's series. 7 Mat. Lapok (1956), 77-100 Hungarian. Russian, English summaries. MR 0099968 | Zbl 0075.03703
[16] Rényi, A.: On a new axiomatic theory of probability. Acta Math. Acad. Sci. Hung. 6 (1955), 285-335. DOI 10.1007/BF02024393 | MR 0081008 | Zbl 0067.10401
[17] Sierpiński, W.: Démonstration élémentaire du théorème de M. Borel sur les nombres absolument normaux et détermination effective d'{u}ne tel nombre. Bull. Soc. Math. Fr. 45 (1917), 125-153 French. MR 1504764
[18] Turán, P.: On the distribution of digits'' in Cantor systems. Mat. Lapok 7 (1956), 71-76 Hungarian. Russian, English summaries. MR 0099967 | Zbl 0075.25202
[19] Turing, A. M.: Collected Works of A. M. Turing: Pure Mathematics. North-Holland Publishing, Amsterdam J. L. Britton (1992). MR 1150052 | Zbl 0751.01017

Partner of