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Indefinite quadratic forms in many variables

Published online by Cambridge University Press:  26 February 2010

H. Davenport
H. Davenport
Affiliation:
University College, London
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Extract

It has long been conjectured that any indefinite quadratic form, with real coefficients, in 5 or more variables assumes values arbitrarily near to 0 for suitable integral values of the variables, not all 0. The basis for this conjecture is the fact, proved by Meyer in 1883, that any such form with rational coefficients actually represents 0.

Type
Research Article
Information
Mathematika , Volume 3 , Issue 2 , December 1956 , pp. 81 - 101
Copyright
Copyright © University College London 1956

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References

page 81 note * See Oppenheim, , Proc. Nat. Acad. Sci. U.S.A., 15 (1929), 724–727.CrossRefGoogle Scholar

page 81 note † J. London Math. Soc., 21 (1946), 185–193.Google Scholar

page 81 note ‡ Proc. London Math. Soc. (3), 3 (1953), 170–181.Google Scholar

page 82 note * Quart. J. of Math. (2), 4 (1953), 54–59CrossRefGoogle Scholar (Theorem 1) and ibid. 60–66 (Theorem 1).

page 82 note † Proc. Royal Soc. (A), 187 (1946), 151–187.

page 83 note * Proc. Cambridge Phil. Soc., 51 (1955), 262–264CrossRefGoogle Scholar and 62 (1956), 604.

page 85 note * We use Vinogradov's symbolism F « G to mean that | F | < cG for a suitable constant c.

page 87 note * Here ‖ θ ‖ denotes the difference between a real number θ and the nearest integer, taken positively.