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A Metrical Theorem in Diophantine Approximation

Published online by Cambridge University Press:  20 November 2018

Wolfgang Schmidt
Wolfgang Schmidt*
Affiliation:
Montana State University
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In this paper we prove a sharpening and generalization of the following Theorem of Khintchine (4):

Let ψ1(q), …, ψnq) be n non-negative junctions of the positive integer q and assume

is monotonically decreasing. Then the set of inequalities

1

has an infinity of integer solutions q > 0 and p1, … , pn for almost all or no sets of numbers θ1, … , θ2, according as Σψ(q) diverges or converges.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 12 , 1960 , pp. 619 - 631
Copyright
Copyright © Canadian Mathematical Society 1960

References

1. Cassels, J.W.S., Some metrical theorems in diophantine approximation I, Proc. Camb. Phil. Soc, 46 (1950), 209–218.Google Scholar
2. Cassels, J.W.S., Some metrical theorems in diophantine approximation III, Proc. Camb. Phil. Soc, 46 (1950), 219–225.Google Scholar
3. Cassels, J.W.S.,An introduction to diophantine approximation, Cambridge Tracts, 45 (1957).Google Scholar
4. Khintchine, A., Zur metrischen Théorie der diophantischen Approximationen, Math. Z., 24 (1926), 706–714.Google Scholar
5. Schmidt, W., A metrical theorem in geometry ojnumbers, Trans. Amer. Math. Soc, 00 (1960), 000–000.Google Scholar