Continued fractions and irrational rotations |
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| Authors: | Naoto Shimaru Keizo Takashima |
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| Affiliation: | 1.Department of Applied Mathematics, Graduate School of Science,Okayama University of Science,Okayama,Japan;2.Department of Applied Mathematics,Okayama University of Science,Okayama,Japan |
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| Abstract: | Let (alpha in (0, 1)) be an irrational number with continued fraction expansion (alpha =[0; a_1, a_2, ldots ]) and let (p_n/q_n= [0; a_1, ldots , a_n]) be the nth convergent to (alpha ). We prove a formula for (p_nq_k-q_np_k) ((k in terms of a Fibonacci type sequence (Q_n) defined in terms of the (a_n) and use it to provide an exact formula for ({nalpha }) for all n. |
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