Investigation of systems of linear diophantine inequalities by group theory methods (Q1278054)

The author considers a system of linear Diophantine inequalities \[ (1)\qquad \langle a^i, x\rangle\leq b_i\quad (i= 1,\dots,\ell),\qquad (2)\qquad x\in\mathbb{Z}^n, \] where \(\mathbb{Z}^n\) is the set of integer-valued vectors in \(\mathbb{R}^n\) and \(a^i\), \(x\) is the scalar product of vectors \(a^i\), \(x\). It is supposed that the set of solutions of system (1) is nonempty and bounded. Since condition (2) is an invariant of the group of motions of the integer lattice \(\mathbb{Z}^n\), system (1), (2) can be investigated by group theory methods. The article contains four theorems representing the results of this investigation and their proofs.