Bull. Korean Math. Soc. 53 (2016), No. 4, pp. 1005–1015 http://dx.doi.org/10.4134/BKMS.b150402 pISSN: 1015-8634 / eISSN: 2234-3016 BOUNDED PARTIAL QUOTIENTS OF SOME CUBIC POWER SERIES WITH BINARY COEFFICIENTS Khalil Ayadi, Salah Beldi, and Kwankyu Lee 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. 1. Introduction Khinchin [4] conjectured that no algebraic real number of degree higher than two has bounded partial quotients in its continued fraction expansion. Then betraying the innocent expectation that the same would be true for algebraic power series, Baum and Sweet [1] found an example of cubic power series over F 2 with partial quotients of bounded degree. Then it was realised, first by Mills and Robbins [11], that irrational power series with coefficients in a finite field F with characteristic p that are roots of equations of the form x = Ax p r + B Cx p r + D , r ≥ 0, A, B, C, D ∈ F[X ] loosely correspond to the quadratic real numbers, and tends to have regular patterns in their continued fraction expansions. These are now called hyper- quadratic power series. Hyperquadratic power series are fixed points under the composition of a linear fractional transformation and a Frobenius map of F. It is known [2, 15] that if a hyperquadratic power series does not have bounded partial quotients, then it has partial quotients of rapidly increasing degrees. So it is either badly approximable or very well approximable by ra- tional functions. Hence the cubic power series of Baum and Sweet belongs to the class of badly approximable hyperquadratic power series over the field F 2 . Received May 27, 2015; Revised April 25, 2016. 2010 Mathematics Subject Classification. 11J61, 11J70. Key words and phrases. power series, continued fraction, finite fields. This work was supported by research fund from Chosun University, 2015. c 2016 Korean Mathematical Society 1005