Hostname: page-component-cd9895bd7-jn8rn Total loading time: 0 Render date: 2024-12-24T13:30:49.995Z Has data issue: false hasContentIssue false
  • English
  • Français

Diophantine approximation with square-free integers

Published online by Cambridge University Press:  24 October 2008

Glyn Harman
Glyn Harman
Affiliation:
Imperial College, London
Get access Rights & Permissions [Opens in a new window]

Extract

In this paper we shall prove the following two results.

Theorem 1. Let ∊ > 0 and β a real number be given. Them, for almost all real a (in the sense of Lebesque measure), there are infinitely many pairs of square-free integers m, n such that

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 95 , Issue 3 , May 1984 , pp. 381 - 388
Copyright
Copyright © Cambridge Philosophical Society 1984

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Davenport, H.. Multiplicative Number Theory, 2nd ed. Revised by Montgomery, H. L.. (Springer-Verlag, 1980).CrossRefGoogle Scholar
[2]Ghosh, A.. The distribution of ap2 modulo one. Proc. London Math. Soc. (3) 42 (1981), 252–69.Google Scholar
[3]Hardy, G. H. and Wright, E. M.. An Introduction to the Theory of Numbers, 5th ed. (Oxford University Press, 1979).Google Scholar
[4]Harman, G.. Trigonometric sums over primes. II. Glasgow Math. J. 24 (1983), 23–37.CrossRefGoogle Scholar
[5]Harman, G.. On the distribution of αp modulo one. J. London Math. Soc. (2) 27 (1983), 9–18.CrossRefGoogle Scholar
[6]Harman, G.. Diophantine approximation with a prime and an almost-prime. J. London. Math. Soc. to appear.Google Scholar
[7]Heath-Brown, D. R.. The least square-free number in an arithmetic progrssion. J. Reine Angew. Math. 332 (1982), 204–220.Google Scholar
[8]Hooley, C.. On a new technique and its application to the theory of numbers. Proc. London Math. Soc. (3) 38 (1979), 115–151.Google Scholar
[9]Iwaniec, H.. On indefinite quadratic forms in four variables. Acta Arith. 33 (1977), 209–229.CrossRefGoogle Scholar
[10]Leveque, W. J.. On the frequency of small fractional pats in certain real sequences. III. J. Reine Angew. Math. 202 (1959), 215–220.Google Scholar
[11]Montgomery, H. L.. Topics in Multiplicative Number Theory. (Springer-Verlag, 1971).CrossRefGoogle Scholar
[12]Sprindzuk, V. G.. Metric Theory of Diophantine Approximations, translated by Silverman, R. A.. (Winston/Wiley, 1979).Google Scholar
[13]Vaughan, R. C.. Diophantine approximation by prime numbers. III. Proc. London Math. Soc., (3) 33 (1976), 177–192.CrossRefGoogle Scholar