On a conjecture of jackson on non-homogeneous quadratic forms

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