Discrepancy estimates for a class of normal numbers

Yoshinobu Nakai; Iekata Shiokawa

Acta Arithmetica (1992)

  • Volume: 62, Issue: 3, page 271-284
  • ISSN: 0065-1036

How to cite


Yoshinobu Nakai, and Iekata Shiokawa. "Discrepancy estimates for a class of normal numbers." Acta Arithmetica 62.3 (1992): 271-284. <http://eudml.org/doc/206493>.

author = {Yoshinobu Nakai, Iekata Shiokawa},
journal = {Acta Arithmetica},
keywords = {normal numbers; real polynomial; Champernowne number; discrepancy; block of digits},
language = {eng},
number = {3},
pages = {271-284},
title = {Discrepancy estimates for a class of normal numbers},
url = {http://eudml.org/doc/206493},
volume = {62},
year = {1992},

AU - Yoshinobu Nakai
AU - Iekata Shiokawa
TI - Discrepancy estimates for a class of normal numbers
JO - Acta Arithmetica
PY - 1992
VL - 62
IS - 3
SP - 271
EP - 284
LA - eng
KW - normal numbers; real polynomial; Champernowne number; discrepancy; block of digits
UR - http://eudml.org/doc/206493
ER -


  1. [1] A. S. Besicovitch, The asymptotic distribution of the numerals in the decimal representation of the squares of the natural numbers, Math. Z. 39 (1935), 146-156. Zbl0009.20002
  2. [2] D. G. Champernowne, The construction of decimals normal in the scale of ten, J. London Math. Soc. 8 (1933), 254-260. Zbl0007.33701
  3. [3] H. Davenport and P. Erdős, Note on normal decimals, Canad. J. Math. 4 (1952), 58-63. Zbl0046.04902
  4. [4] L.-K. Hua, Additive Theory of Prime Numbers, Transl. Math. Monographs 13, Amer. Math. Soc., Providence, Rhode Island, 1965. Zbl0192.39304
  5. [5] A. A. Karatsuba, Foundations of Analytic Theory of Numbers, 2nd ed., Nauka, 1983 (in Russian). 
  6. [6] L. Kuipers and H. Niederreiter, Uniform Distribution of Sequences, Wiley, New York 1974. Zbl0281.10001
  7. [7] L. Mirsky, A theorem on representations of integers in the scale of r, Scripta Math. 15 (1974), 11-12. Zbl0034.17102
  8. [8] Y.-N. Nakai and I. Shiokawa, A class of normal numbers, Japan. J. Math. 16 (1990), 17-29. 
  9. [9] Y.-N. Nakai and I. Shiokawa, A class of normal numbers II, in: Number Theory and Cryptography, J. H. Loxton (ed.), London Math. Soc. Lecture Note Ser. 154, Cambridge Univ. Press, 1990, 204-210. 
  10. [10] J. Schiffer, Discrepancy of normal numbers, Acta Arith. 47 (1986), 175-186. Zbl0556.10036
  11. [11] J. Schoißengeier, Über die Diskrepanz von Folgen (αbⁿ), Sitzungsber. Öst. Akad. Wiss. Math.-Natur. Kl. Abt. II 187 (1978), 225-235. 
  12. [12] E. C. Titchmarsh, The Theory of the Riemann Zeta-Function, Oxford Univ. Press, 1951. Zbl0042.07901
  13. [13] R. C. Vaughan, The Hardy-Littlewood Method, Cambridge Tracts in Math. 80, Cambridge Univ. Press, London 1981. 
  14. [14] I. M. Vinogradov, The Method of Trigonometric Sums in Number Theory, Nauka, 1971 (in Russian). Zbl0229.10020

NotesEmbed ?


You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.


Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.