On a conjecture of jackson on non-homogeneous quadratic forms

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(x1,…, xn) be a real indefinite quadratic form of determinant D ≠ 0. Let ||α||≤ ||D||1/n. For any real numbers a1,…, an, there exist (x1,…, xn) = (a1,…, an) (mod 1) such that |Q(x1.....xn) - α|≤|D|1/n. In particular, the proof shows that we can find (x1,…, xn) = (a1,…, an) (mod 1) such that 0 < Q(x1 . . . . .xn) ≤ 2|D|1/n. For forms of signature zero this result is also the best possible.

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