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Equidistribution of values of rational functions (mod p)

Published online by Cambridge University Press:  14 November 2011

R. W. K. Odoni  and
P. G. Spain
R. W. K. Odoni
Affiliation:
Mathematics Department, University of Glasgow, Glasgow G12 8QW, Scotland, U.K
P. G. Spain
Affiliation:
Mathematics Department, University of Glasgow, Glasgow G12 8QW, Scotland, U.K.; Institute of Mathematics, Hebrew University of Jerusalem, 91904 Jerusalem, Israel
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Extract

Let R1(x),…, Rd(x) be rational functions in Iℚ(x), such that 1, R1(x),…, Rd(x) are linearly independent over Iℚ. For almost all primes p, their mod p reductions, are well-defined rational functions over Fp and are linearly independent over Fp We show that asymptotically the points

are uniformly distributed in [0, l)d.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 125 , Issue 5 , 1995 , pp. 911 - 929
Copyright
Copyright © Royal Society of Edinburgh 1995

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References

1Davenport, H.. On a principle of Lipschitz. J. London Math. Soc. 26 (1951), 179–83.CrossRefGoogle Scholar
2Deligne, P.. La conjecture de Weil—I. Inst. Hautes Études Sci. Publ. Math. 43 (1974), 273–307.CrossRefGoogle Scholar
3Dym, H. and McKean, H. P.. Fourier Series and Integrals (New York: Academic Press, 1972).Google Scholar
4Erdös, P.. Some remarks on Diophantine Approximations. J. Indian Math. Soc. 12 (1948), 67–74.Google Scholar
5Hardy, G. H. and Littlewood, J. E.. Some problems of Diophantine approximation: The lattice-points of a right-angled triangle. Proc. London Math. Soc. (2) 20 (1921), 15–36.Google Scholar
6Hardy, G. H. and Littlewood, J. E.. The lattice-points of a right-angled triangle. (Second memoir) Abh. Math. Sem. Univ. Hamburg 1 (1992), 212–49.Google Scholar
7Hartshorne, R.. Algebraic Geometry (Berlin: Springer, 1983).Google Scholar
8Katz, N. M.. Sommes exponentielles. astérisque 79 (1980).Google Scholar
9Kruse, A. H.. Estimates of . Acta Arith. 12 (1966–1967), 229–61.CrossRefGoogle Scholar
10Lang, S. and Weil, A.. Number of points of varieties in finite fields. Amer. J. Math. 76 (1954), 819–27.CrossRefGoogle Scholar
11Lipschitz, R.. Monatsber. der Berliner Academie (1865), 174 et seq.Google Scholar
12Weil, A.. On the Riemann hypothesis in function-fields. Proc. Nat. Acad. Sci. U.S.A. 27 (1941), 345–7.CrossRefGoogle ScholarPubMed
13Weil, A.. Basic Number Theory, 3rd edn (Berlin: Springer, 1974).CrossRefGoogle Scholar
14Zygmund, A.. Trigonometric Series (Edinburgh: Cambridge University Press, 1968).Google Scholar