$\mathcal{L}$-algorithm for approximating Diophantine systems of linear forms

It is proposed $\mathcal{L}$ - algorithm for constructing an infinite sequence of integer solutions of linear inequality systems of $ d + 1 $ variable. Solutions are obtained using recurrence relations of order $d + 1$. The approach speed is carried out with the diophantine exponent $\theta = \frac {m} {n} - \varrho $ where $ 1 \leq n \leq d $ is the number of inequalities, $ m = d + 1-n $ — the number of free variables and the deviation $ \varrho> 0 $ can be made arbitrarily small due to a suitable choice of the recurrence relation.