zbMATH — the first resource for mathematics

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)].

11J20 Inhomogeneous linear forms
11J70 Continued fractions and generalizations
Full Text: DOI EMIS Numdam EuDML
[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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.