BOUNDED PARTIAL QUOTIENTS OF SOME CUBIC POWER SERIES WITH BINARY COEFFICIENTS

- Journal title : Bulletin of the Korean Mathematical Society
- Volume 53, Issue 4, 2016, pp.1005-1015
- Publisher : The Korean Mathematical Society
- DOI : 10.4134/BKMS.b150402

Title & Authors

BOUNDED PARTIAL QUOTIENTS OF SOME CUBIC POWER SERIES WITH BINARY COEFFICIENTS

Ayadi, Khalil; Beldi, Salah; Lee, Kwankyu;

Ayadi, Khalil; Beldi, Salah; Lee, Kwankyu;

Abstract

It is a surprising but now well-known fact that there exist algebraic power series of degree higher than two with partial quotients of bounded degrees in their continued fraction expansions, while there is no single algebraic real number known with bounded partial quotients. However, it seems that these special algebraic power series are quite rare and it is hard to determine their continued fraction expansions explicitly. To the short list of known examples, we add a new family of cubic power series with bounded partial quotients.

Keywords

power series;continued fraction;finite fields;

Language

English

References

1.

L. E. Baum and M. M. Sweet, Continued fractions of algebraic power series in characteristic 2, Ann. of Math. (2) 103 (1976), no. 3, 593-610.

2.

B. de Mathan, Approximation exponents for algebraic functions in positive characteristic, Acta Arith. 60 (1992), no. 4, 359-370.

3.

D. Gomez-Perez and A. Lasjaunias, Hyperquadratic power series in $F_{3}((T^{-1}))$ with partial quotients of degree 1, Ramanujan J. 33 (2014), no. 2, 219-226.

4.

A. I. Khinchin, Continued Fractions, University of Chicago Press, 1964.

5.

A. Lasjaunias, Continued fractions for algebraic formal power series over a finite base field, Finite Fields Appl. 5 (1999), no. 1, 46-56.

6.

A. Lasjaunias, Quartic power series in $F_{3}((T^{-1}))$ with bounded partial quotients, Acta Arith. 95 (2000), no. 1, 49-59.

7.

A. Lasjaunias and J.-J. Ruch, Algebraic and badly approximable power series over a finite field, Finite Fields Appl. 8 (2002), no. 1, 91-107.

8.

A. Lasjaunias and J.-J. Ruch, Flat power series over a finite field, J. Number Theory 95 (2002), no. 2, 268-288.

9.

A. Lasjaunias and J.-Y. Yao, Hyperquadratic continued fractions in odd characteristic with partial quotients of degree one, J. Number Theory 149 (2015), 259-284.

10.

K. Lee, Continued fractions for linear fractional transformations of power series, Finite Fields Appl. 11 (2005), no. 1, 45-55.

11.

W. H. Mills and D. P. Robbins, Continued fractions for certain algebraic power series, J. Number Theory 23 (1986), no. 3, 388-404.

12.

M. Mkaouar, Sur les fractions continues des series formelles quadratiques sur $f_q(x)$ , Acta Arith. 97 (2001), no. 3, 241-251.

13.

W. M. Schmidt, On continued fractions and diophantine approximation in power series fields, Acta Arith. 95 (2000), no. 2, 139-166.