Dual Diophantine Systems of Linear Inequalities

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.