Explicit lower bounds for linear forms

Abstract

Let I \mathbb {I} be the field of rational numbers or an imaginary quadratic field and Z I \mathbb {Z}_\mathbb {I} its ring of integers. We study some general lemmas that produce lower bounds \[ | B 0 + B 1 θ 1 + ⋯ + B r θ r | ≥ 1 max { | B 1 | , … , | B r | } μ \lvert B_0+B_1\theta _1+\cdots +B_r\theta _r\rvert \ge \frac {1}{\max \{\lvert B_1 \rvert ,\ldots ,\lvert B_r \rvert \}^\mu } \] for all B 0 , … , B r ∈ Z I B_0,\ldots ,B_r \in \mathbb {Z}_{\mathbb {I}} , max { | B 1 | , … , | B r | } ≥ H 0 \max \{\lvert B_1 \rvert ,\ldots ,\lvert B_r \rvert \} \ge H_0 , given suitable simultaneous approximating sequences of the numbers θ 1 , … , θ r \theta _1,\ldots ,\theta _r . We manage to replace the lower bound with 1 / max { | B 1 | μ 1 , … , | B r | μ r } 1/{\max \{\lvert B_1 \rvert ^{\mu _1},\ldots ,\lvert B_r \rvert ^{\mu _r}\}} for all B 0 , … , B r ∈ Z I B_0,\ldots ,B_r \in \mathbb {Z}_{\mathbb {I}} , max { | B 1 | μ 1 , … , | B r | μ r } ≥ H 0 \max \{\lvert B_1 \rvert ^{\mu _1},\ldots ,\lvert B_r \rvert ^{\mu _r}\} \ge H_0 , where the exponents μ 1 , … , μ r \mu _1,\ldots ,\mu _r are different when the given type II approximating sequences approximate some of the numbers θ 1 , … , θ r \theta _1,\ldots ,\theta _r better than the others. As an application we research certain linear forms in logarithms. Our results are completely explicit.