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Diophantine approximations with Fibonacci numbers. (English. French summary) Zbl 1283.11102

Summary: Let \(F_n\) be the \(n\)th Fibonacci number. Put \(\varphi =\frac{1+\sqrt 5}2\). The author proves that the following inequalities hold for any real \(\alpha \):
1) \(\inf_{n\in \mathbb N } ||F_n\alpha ||\leq \frac{\varphi-1}{\varphi+2}\),
2) \(\liminf_{ n\to\infty }||F_n\alpha ||\leq \frac 15\),
3) \(\liminf_{n\to\infty}||\varphi^n\alpha ||\leq \frac 15\).
These bounds are sharp.

MSC:

11J25 Diophantine inequalities
11B39 Fibonacci and Lucas numbers and polynomials and generalizations
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References:

[1] R. K. Akhunzhanov, On the distribution modulo 1 of exponential sequences. Mathematical notes 76:2 (2004), 153-160. · Zbl 1196.11107
[2] A. Dubickas, Arithmetical properties of powers of algebraic numbers. Bull. London Math. Soc. 38 (2006), 70-80. · Zbl 1164.11025
[3] L. Kuipers, H. Niederreiter, Uniform distribution of sequences. John Wiley & Sons, 1974. · Zbl 0281.10001
[4] W. M. Schmidt, Diophantine approximations. Lect. Not. Math. 785, 1980. · Zbl 0421.10019
[5] W. M. Schmidt, On badly approximable numbers and certain games. Trans. Amer. Math. Soc. 623 (1966), 178-199. · Zbl 0232.10029
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