TY - JOUR
T1 - On the equivalence relations of α-continued fractions
AU - Nakada, Hitoshi
AU - Natsui, Rie
N1 - Funding Information:
The authors would like to thank the referee for careful reading of an early version of this paper and valuable comments. This research was supported in part by the Grant-in-Aid for Scientific research (Nos. 24340020 and 23740088 ), the Japan Society for the Promotion of Science .
PY - 2014/6/27
Y1 - 2014/6/27
N2 - We compare the equivalence relations on real numbers arising from having eventually agreeing α-continued fraction expansion (for each 0 ≤ α ≤ 1) with those from sharing GL(2,Z)- or SL(2,Z)-orbits. We show that the α-relation and the GL(2,Z)-relation of x are identical for any α > 0 when x is of unbounded type. On the other hand, they are not identical for x of bounded type.
AB - We compare the equivalence relations on real numbers arising from having eventually agreeing α-continued fraction expansion (for each 0 ≤ α ≤ 1) with those from sharing GL(2,Z)- or SL(2,Z)-orbits. We show that the α-relation and the GL(2,Z)-relation of x are identical for any α > 0 when x is of unbounded type. On the other hand, they are not identical for x of bounded type.
KW - Continued fractions
KW - Diophantine approximations
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U2 - 10.1016/j.indag.2014.02.006
DO - 10.1016/j.indag.2014.02.006
M3 - Article
AN - SCOPUS:84902330874
SN - 0019-3577
VL - 25
SP - 800
EP - 815
JO - Indagationes Mathematicae
JF - Indagationes Mathematicae
IS - 4
ER -