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Numbers badly approximate by fractions with prime denominator

Published online by Cambridge University Press:  24 October 2008

Glyn Harman
Glyn Harman
Affiliation:
School of Mathematics, University of Wales College of Cardiff, Senghennydd Road, Cardiff CF2 4AG
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Extract

We write ‖x‖ to denote the least distance from x to an integer, and write p for a prime variable. Duffin and Schaeffer [l] showed that for almost all real α the inequality

has infinitely many solutions if and only if

diverges. Thus f(x) = (x log log (10x))−1 is a suitable choice to obtain infinitely many solutions for almost all α. It has been shown [2] that for all real irrational α there are infinitely many solutions to (1) with f(p) = p−/13. We will show elsewhere that the exponent can be increased to 7/22. A very strong result on primes in arithmetic progressions (far stronger than anything within reach at the present time) would lead to an improvement on this result. On the other hand, it is very easy to find irrational a such that no convergent to its continued fraction expansion has prime denominator (for example (45– √10)/186 does not even have a square-free denominator in its continued fraction expansion, since the denominators are alternately divisible by 4 and 9).

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 118 , Issue 1 , July 1995 , pp. 1 - 5
Copyright
Copyright © Cambridge Philosophical Society 1995

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References

REFERENCES

[1]Duffin, R. J. and Schaeffer, A. C.. Khintchine's problem in metric Diophantine approximation. Duke Math. J. 8 (1941), 243–255.Google Scholar
[2]Chao-Hua, Jia. On the distribution of αp modulo one. J. Number Theory 45 (1993), 241–253.Google Scholar
[3]Halberstam, H. and Richert, H.-R.. Sieve methods (Academic Press 1974).Google Scholar
[4]Hardy, G. H. and Wright, E. M.. Introduction to the theory of numbers, fifth edition (Oxford, 1979).Google Scholar
[5]Harman, G.. Metric Diophantine approximation with two restricted variables III. J. Number Theory 29 (1988), 364–375.Google Scholar
[6]Montgomery, H. L. and Vaughan, R. C.. On the large sieve. Mathematika 20 (1973) 119–134.CrossRefGoogle Scholar