Dual Diophantine Systems of Linear Inequalities

Abstract

The paper suggests a modified version of the ℒ-algorithm for constructing an infinite sequence of integral solutions of dual systems $$ \mathcal{S} $$ and $$ {\mathcal{S}}^{\ast } $$ of linear inequalities in d + 1 variables consisting of k⊥ and k*⊥ inequalities, respectively, where k⊥ + k*⊥ = d + 1. Solutions are obtained from two recurrence relations of order d + 1. Approximation in the inequality systems $$ \mathcal{S} $$ and $$ {\mathcal{S}}^{\ast } $$ is effected with the Diophantine exponents $$ \frac{d+1-{k}^{\perp }}{k^{\perp }}-\upvarrho $$ and $$ \frac{d+1-{k}^{\ast \perp }}{k^{\ast \perp }}-\upvarrho $$ , respectively, where the deviation ϱ > 0 can be made arbitrarily small by appropriately choosing the recurrence relations. The ℒ-algorithm is based on a method for localizing units in algebraic number fields.