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More on inhomogeneous Diophantine approximation. (English) Zbl 1014.11043
Given irrational real numbers \(\alpha\) and a real number \(\gamma\) in the unit interval, define the one-sided approximation constants \[ M_+ (\alpha,\gamma)= \liminf_{n\to-\infty} n\|n\alpha- \gamma\|, \qquad M_-(\alpha,\gamma)= \liminf_{n\to_\infty} n\|n\alpha-\gamma\|, \] and the inhomogeneous function \(M(\alpha,\gamma)= \min\{M_+ (\alpha,\gamma), M_- (\alpha,\gamma)\}\), where \(\|x\|\) denotes the distance from \(x\) to the nearest integer. Given any (fixed) irrational \(\alpha\), one is interested in the spectrum \({\mathcal S}(\alpha)\) of values \(M(\alpha,\gamma)\) where \(\gamma\) runs through the real numbers not of the form \(u+\alpha v\) \((u,v\in \mathbb{Z})\). Similarly one defines spectra \({\mathcal S}_+ (\alpha)\), \({\mathcal S}_-(\alpha)\) involving the functions \(M_+\) and \(M_-\), respectively. The author studies these spectra for various classes of numbers \(\alpha\). He works with the backward continued fraction expansion obtained by iterating the singular shift \(T:x\mapsto \lceil 1/x\rceil- 1/x\), and with a kind of \(\alpha\)-expansion of \(\gamma\) [for related work, see T. Komatsu, Acta Arith. 86, 305-324 (1998; Zbl 0930.11049) and T. W. Cusick, A. M. Rockett and P. Szüsz, J. Number Theory 48, 259-283 (1994; Zbl 0820.11042)].

MSC:
11J20 Inhomogeneous linear forms
11J70 Continued fractions and generalizations
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[1] Barnes, E.S., Swinnerton-Dyer, H.P.F., The inhomogeneous minima of binary quadratic forms. Part I, Acta Math.87 (1952), 259-323; Part II, Acta Math.88 (1952), 279-316; Part III, Acta Math.92 (1954), 199-234; Part IV (without second author) Acta Math.92 (1954), 235-264. · Zbl 0056.27301
[2] Cusick, T.W., Rockett, A.M., Szúsz, P., On inhomogeneous Diophantine approximation. J. Number Theory48 (1994), 259-283. · Zbl 0820.11042
[3] Davenport, H., Non-homogeneous binary quadratic forms. Nederl. Akad. Wetensch. Proc.50 (1947), 741-749, 909-917 = Indagationes Math.9 (1947), 351-359, 420-428. · Zbl 0060.11906
[4] Komatsu, T., On inhomogeneous diophantine approximation and the Nishioka - Shiokawa- Tamura algorithm. Acta Arith.86 (1998), 305-324. · Zbl 0930.11049
[5] Moran, W., Pinner, C., Pollington, A., On inhomogeneous Diophantine approximation, preprint.
[6] Varnavides, P., Non-homogeneous quadratic forms, I, II. Nederl. Akad. Wetensch. Proc.51, (1948) 396-404, 470-481. = Indagationes Math.10 (1948), 142-150, 164-175. · Zbl 0030.01901
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