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( 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)−a|⩽|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.