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Diophantine approximation with almost-primes and sums of two squares

Published online by Cambridge University Press:  26 February 2010

Glyn Harman
Affiliation:
Department of Pure Mathematics, University College, P.O. Box 78, Cardiff, CF1 1XL
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Extract

Suppose that θ is a positive irrational number and α is an arbitrary real number. Then Kronecker's Theorem in diophantine approximation (Theorem 440 of [7[) can be stated as follows.

Type
Research Article
Copyright
Copyright © University College London 1985

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References

1.Chen, J.R.. On the representation of a large even integer as the sum of a prime and the product of at most two primes. Sci. Sinica, 16 (1973), 157176.Google Scholar
2.Davenport, H.. Multiplicative Number Theory, 2nd ed. revised by H. L. Montgomery (Springer, New York, 1980).CrossRefGoogle Scholar
3.Elliott, P. D. T. A. and Halberstam, H.. Some applications of Bombieri's theorem. Mathematika, 13 (1967), 196203.CrossRefGoogle Scholar
4.Ghosh, A.. The distribution of α 2 modulo one. Proc. London Math. Soc. (3), 42 (1981), 252269.CrossRefGoogle Scholar
5.Greaves, G.. On the representation of a number in the form x2 + y2 + p2+q2 where p2, q2 are odd primes. Acta Arithmetica, 29 (1976), 257273.CrossRefGoogle Scholar
6.Halberstam, H. and Richert, H. E.. Sieve Methods (Academic Press, London, 1974).Google Scholar
7.Hardy, G. H. and Wright, E. M.. An Introduction to the Theory of Numbers, 5th ed. (Oxford University Press, 1979).Google Scholar
8.Harman, G.. Trigonometric sums over primes II. Glasgow Math. J., 24 (1983), 2337.CrossRefGoogle Scholar
9.Harman, G.. On the distribution of ap modulo one. J. London Math. Soc. (2), 27 (1983), 918.CrossRefGoogle Scholar
10.Harman, G.. Diophantine approximation with a prime and an almost-prime. J. London Math. Soc. (2), 29 (1984), 1322.CrossRefGoogle Scholar
11.Harman, G.. Diophantine approximation with square-free integers. Math. Proc. Camb. Phil Soc. 95 (1984) 381388.CrossRefGoogle Scholar
12.Heath-Brown, D. R.. Three primes and an almost-prime in arithmetic progression. J. London Math. Soc. (2), 23 (1981), 396414.CrossRefGoogle Scholar
13.Heath-Brown, D. R.. Diophantine approximation with square-free integers. Math. Zeit., 187 (1984), 335344.CrossRefGoogle Scholar
14.Hooley, C.. On a new technique and its application to the theory of numbers. Proc. London Math. Soc. (3), 38 (1979), 115151.CrossRefGoogle Scholar
15.Huxley, M. N. and Iwaniec, H.. Bombieri's theorem in short intervals. Mathematika, 22 (1975), 188194.CrossRefGoogle Scholar
16.Iwaniec, H.. Primes of the type ϕ (x, y) + A where ϕ is a quadratic form. Acta Arithmetica, 21 (1972), 203234.CrossRefGoogle Scholar
17.Iwaniec, H.. The half dimensional sieve. Acta Arithmetica, 29 (1976), 6995.CrossRefGoogle Scholar
18.Iwaniec, H.. On indefinite quadratic forms in four variables. Acta Arithmetica, 33 (1977), 209229.CrossRefGoogle Scholar
19.Iwaniec, H.. Rosser's sieve. Acta Arithmetica, 36 (1980), 171202.CrossRefGoogle Scholar
20.Iwaniec, H.. A new form of the error term in the linear sieve. Acta Arithmetica, 37 (1980), 307320.CrossRefGoogle Scholar
21.Landau, E.. Handbuch der Lehre von der Verteilung der Primzahlen (Teubner, Leipzig 1909 or Chelsea, New York, 1953).Google Scholar
22.Vaughan, R. C.. Diophantine approximation by prime numbers III. Proc. London Math. Soc (3), 33 (1976), 177192.CrossRefGoogle Scholar