Bambah, R. P. ; Dumir, V. C. ; Hans-Gill, R. J.
(1983)
*On a conjecture of jackson on non-homogeneous quadratic forms*
Journal of Number Theory, 16
(3).
pp. 403-419.
ISSN 0022-314X

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Official URL: http://linkinghub.elsevier.com/retrieve/pii/002231...

Related URL: http://dx.doi.org/10.1016/0022-314X(83)90067-7

## Abstract

Here we prove the following modification of a conjecture of Jackson (J. London Math. Soc. (2) 3 (1971), 47-58) for indefinite quadratic forms of signature 0, ± 1 or ±2. Let Q(x_{1},…, x_{n}) be a real indefinite quadratic form of determinant D ≠ 0. Let ||α||≤ ||D||^{1/n}. For any real numbers a_{1},…, a_{n}, there exist (x_{1},…, x_{n}) = (a_{1},…, a_{n}) (mod 1) such that |Q(x_{1}.....x_{n}) - α|≤|D|^{1/n}. In particular, the proof shows that we can find (x_{1},…, x_{n}) = (a_{1},…, a_{n}) (mod 1) such that 0 < Q(x_{1} . . . . .x_{n}) ≤ 2|D|^{1/n}. For forms of signature zero this result is also the best possible.

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ID Code: | 12349 |

Deposited On: | 10 Nov 2010 06:25 |

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