A quantitative version of the Absolute Subspace Theorem

The celebrated Subspace Theorem of W. M. Schmidt [12] says the following: SUBSPACE THEOREM. Let L1, . . . , Ln be linearly independent linear forms in n variables, with real or complex algebraic coefficients. Suppose δ > 0. Consider the inequality (1.1) |L1(x) . . . Ln(x)| 0, put (1.5) λΠ(Q, c) = {x ∈ R | |Li(x)| ≤ Qiλ (1 ≤ i ≤ n)} . Then the Parametric Subspace Theorem can be stated as follows. PARAMETRIC SUBSPACE THEOREM. Let c be a fixed tuple satisfying (1.3). Let δ > 0. Then there are finitely many proper linear subspaces T1, . . . , TB of Q n such that for any Q which is sufficiently large there exists a subspace Ti ∈ {T1, . . . , TB} with (1.6) Q−δΠ(Q, c) ∩ Z ⊂ Ti .