New bounds on the length of finite pierce and Engel series

P. Erdös; J. O. Shallit

Journal de théorie des nombres de Bordeaux (1991)

  • Volume: 3, Issue: 1, page 43-53
  • ISSN: 1246-7405

Abstract

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Every real number x , 0 < x ≤ 1 , has an essentially unique expansion as a Pierce series : x = 1 x 1 - 1 x 1 x 2 + 1 x 1 x 2 x 3 - ⋯ where the x i form a strictly increasing sequence of positive integers. The expansion terminates if and only if x is rational. Similarly, every positive real number y has a unique expansion as an Engel series : y = 1 y 1 - 1 y 1 y 2 + 1 y 1 y 2 y 3 + ⋯ where the y i form a (not necessarily strictly) increasing sequence of positive integers. If the expansion is infinite, we require that the sequence yi be not eventually constant. Again, such an expansion terminates if and only if y is rational. In this paper we obtain some new upper and lower bounds on the lengths of these series on rational inputs a / b . In the case of the Engel series, this answers an open question of Erdös, Rényi, and Szüsz. However, our upper and lower bounds are widely separated.

How to cite

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Erdös, P., and Shallit, J. O.. "New bounds on the length of finite pierce and Engel series." Journal de théorie des nombres de Bordeaux 3.1 (1991): 43-53. <http://eudml.org/doc/93535>.

@article{Erdös1991,
abstract = {Every real number $x, 0 &lt; x \le 1$, has an essentially unique expansion as a Pierce series :\begin\{equation*\} x = \frac\{1\}\{x^1\}- \frac\{1\}\{x^1 x^2\} + \frac\{1\}\{x^1 x^2 x^3\} - \cdots \end\{equation*\}where the $x_i$ form a strictly increasing sequence of positive integers. The expansion terminates if and only if $x$ is rational. Similarly, every positive real number $y$ has a unique expansion as an Engel series :\begin\{equation*\} y = \frac\{1\}\{y^1\}- \frac\{1\}\{y^1 y^2\} + \frac\{1\}\{y^1 y^2 y^3\} + \cdots \end\{equation*\}where the $y_i$ form a (not necessarily strictly) increasing sequence of positive integers. If the expansion is infinite, we require that the sequence yi be not eventually constant. Again, such an expansion terminates if and only if $y$ is rational. In this paper we obtain some new upper and lower bounds on the lengths of these series on rational inputs $a/b$. In the case of the Engel series, this answers an open question of Erdös, Rényi, and Szüsz. However, our upper and lower bounds are widely separated.},
author = {Erdös, P., Shallit, J. O.},
journal = {Journal de théorie des nombres de Bordeaux},
keywords = {Pierce series; Engel series; rational numbers; length of Engel series expansions; alternating Engel series; Pierce expansions},
language = {eng},
number = {1},
pages = {43-53},
publisher = {Université Bordeaux I},
title = {New bounds on the length of finite pierce and Engel series},
url = {http://eudml.org/doc/93535},
volume = {3},
year = {1991},
}

TY - JOUR
AU - Erdös, P.
AU - Shallit, J. O.
TI - New bounds on the length of finite pierce and Engel series
JO - Journal de théorie des nombres de Bordeaux
PY - 1991
PB - Université Bordeaux I
VL - 3
IS - 1
SP - 43
EP - 53
AB - Every real number $x, 0 &lt; x \le 1$, has an essentially unique expansion as a Pierce series :\begin{equation*} x = \frac{1}{x^1}- \frac{1}{x^1 x^2} + \frac{1}{x^1 x^2 x^3} - \cdots \end{equation*}where the $x_i$ form a strictly increasing sequence of positive integers. The expansion terminates if and only if $x$ is rational. Similarly, every positive real number $y$ has a unique expansion as an Engel series :\begin{equation*} y = \frac{1}{y^1}- \frac{1}{y^1 y^2} + \frac{1}{y^1 y^2 y^3} + \cdots \end{equation*}where the $y_i$ form a (not necessarily strictly) increasing sequence of positive integers. If the expansion is infinite, we require that the sequence yi be not eventually constant. Again, such an expansion terminates if and only if $y$ is rational. In this paper we obtain some new upper and lower bounds on the lengths of these series on rational inputs $a/b$. In the case of the Engel series, this answers an open question of Erdös, Rényi, and Szüsz. However, our upper and lower bounds are widely separated.
LA - eng
KW - Pierce series; Engel series; rational numbers; length of Engel series expansions; alternating Engel series; Pierce expansions
UR - http://eudml.org/doc/93535
ER -

References

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  4. 4 P. Erdös, A. Rényi, and P. Szüsz, On Engel's and Sylvester's series, Ann. Univ. Sci. Budapest. Eötvös Sect. Math.1 (1958), 7-32. Zbl0107.27002MR102496
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  8. 8 E. Ya. Remez, On series with alternating signs which may be connected with two algorithms of M. V. Ostrogradskii for the approximation of irrational numbers, Uspekhi Mat. Nauk6 (5) (1951), 33-42, (MR #13,444d). Zbl0045.02102MR44585
  9. 9 A. Rényi, A new approach to the theory of Engel's series, Ann. Univ. Sci. Budapest. Eötvös Sect. Math.5 (1962), 25-32. Zbl0232.10028MR150123
  10. 10 J.B. Rosser and L. Schoenfeld, Approxamate formulas for some functions of prime numbers, Illinois J. Math.6 (1962), 64-94. Zbl0122.05001MR137689
  11. 11 J.O. Shallit, Metric theory of Pierce expansions, Fibonacci Quart.24 (1986), 22-40. Zbl0598.10057MR825872
  12. 12 J.O. Shallit, Letter to the editor, Fibonacci Quart.27 (1989), 186. 
  13. 13 W. Sierpinski, O kilku algorytmach dla rozwijania liczb rzeczywistych na szeregi, C. R. Soc. Sci. Varsovie4 (1911), 56-77, (In Polish; reprinted in French translation as Sur quelques algorithmes pour développer les nombres reéls en séries, in W. Sierpinski, Oeuvres Choisies, Vol. I, PWN, Warsaw, 1974, pp. 236-254.). 
  14. 14 K.G. Valeyev and E.D. Zlebov, The metric theory of an algorithm of M. V. Ostrogradskij, Ukrain. Mat. Z.27 (1975), 64-69. Zbl0309.10020MR366855

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