Simultaneous diophantine approximation with quadratic and linear forms

Abstract

Let Q be a nondegenerate indefinite quadratic form on Rⁿ, n≥3, which is not a scalar multiple of a rational quadratic form, and let C_Q={v∈Rⁿ|Q(v)=0}. We show that given v₁∈C_Q, for almost all v∈C_Q\Rv₁ the following holds: for any a∈R, any affine plane P parallel to the plane of v₁ and v, and ∈>0 there exist primitive integral n-tuples x within ∈ distance of P for which |Q(x)-a|<∈. An analogous result is also proved for almost all lines on C_Q.