On a conjecture of Jackson on non-homogeneous quadratic forms

AbstractHere 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∥1n. For any real numbers a1,…, an, there exist (x1,…, xn) ≡ (a1,…, an) (mod 1) such that |Q(x1,…,xn)−a|⩽|D|1n. In particular, the proof shows that we can find (x1,…, xn) ≡ (a1,…, an) (mod 1) such that 0 < Q(x1,…xn)⩽2|D|1n For forms of signature zero this result is also the best possible