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Indefinite quadratic polynomials

Published online by Cambridge University Press:  26 February 2010

G. L. Watson
G. L. Watson
Affiliation:
University College, London, W.C.I.
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1. Let f = f(x1, …, xn) be an indefinite quadratic form in n variables with discriminant d = d(f) ¹ 0; and let ξ1, …, ξn be real numbers. We consider how closely the inhomogeneous quadratic polynomial

can be made to approximate to a given real number α by choice of suitable integral values of the variables xi. The best that is known seems to be that the inequalities

can always be satisfied if the implied constant is given a suitable value depending only on n. For α ≥ 0 this is a restatement of a result proved by Dr. D. M. E. Foster.

Type
Research Article
Information
Mathematika , Volume 7 , Issue 2 , December 1960 , pp. 141 - 144
Copyright
Copyright © University College London 1960

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References

page 141 note * Mathematika, 3 (1956), 111–116CrossRefGoogle Scholar, Theorem 1.

page 141 note † See Ridout, D., Mathematika, 5 (1958), 122–124CrossRefGoogle Scholar, for references and proof of one outstanding case.

page142 note * Oppenheim, A., Monatshefte für Mathematik, 57 (1954), 97–101CrossRefGoogle Scholar; the writer outlines an alternative proof found independently by Heilbronn, which appears not to have been published.

page142 note † Oppenheim, A., Quart. J. Math. Oxford (2), 4 (1953), 54–59 (54, Theorem 1).CrossRefGoogle Scholar

page143 note * Diophantische Approximationen (Ergebnisse der Mathematik und ihrer Grenzgebiete) (New York), 93, Satz 10.Google Scholar