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The lattice properties of asymmetric hyperbolic regions

I. On a theorem of khintchine

Published online by Cambridge University Press:  24 October 2008

J. W. S. Cassels
J. W. S. Cassels
Affiliation:
Trinity CollegeCambridge
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Extract

Let θ > 0 and α ≠ 0 be real numbers, and let θ be irrational. Khintchine has shown, by the use of continued fractions, that there is an infinite number of pairs of positive integers (p, q) which satisfy the inequality

for any given K > 5−½; and, more recently, Jogin has shown the same is still true with K = 5−½. The condition that p and q shall be positive is, of course, essential, as otherwise there is the classical result K = ¼ due to Minkowski.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 44 , Issue 1 , January 1948 , pp. 1 - 7
Copyright
Copyright © Cambridge Philosophical Society 1948

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References

* Khintchine, A., Math. Ann. 111 (1935), 631–7.CrossRefGoogle Scholar

† Jogin, I. I., Uchenye Zapiski Moskov Gos. Univ. Matematika, 73 (1944), 37–40;Google Scholar as quoted in Math. Rev. 7 (1946), 274.Google Scholar

‡ This has recently been improved by Khintchine, , Bull. Acad. Sci. U.R.S.S. 10 (1946), 281–93.Google Scholar Khintchine's results are easily proved by my present methods, and in a subsequent paper of this series I shall show how they can be both generalized and sharpened.

* Hurwitz, , Math. Ann. 39 (1891), 279–81.CrossRefGoogle Scholar See also Hardy, G. H. and Wright, E. M., An Introduction to the Theory of Numbers (Oxford, First edition, 1938; Second edition, 1945),Google Scholar § 11·8. We shall quote this book as Hardy-Wright.

† Hardy-Wright, § 11·10. Koksma, J. F., ‘Diophantische Approximationen’, Ergebn. Math. 4, no. 4 (Springer, Berlin, 1936), pp. 29Google Scholar et.seq.

* A. Khintchine, loc. cit.

† Segre, B., Duke Math. J. 12 (1945), 337–65.CrossRefGoogle Scholar An example is

for integral n.

* O(0, 0) is not an inner point of

† See diagram. is bounded by the thick line.