On linear and multiplicative relations

A theorem on the successive minima of lattices corresponding to the integer solutions of systems of linear equations is proved. As a corollary, theorems on the successive minima are obtained for the set of solutions of equations of the form
$$ x_1\ln\alpha_1+\dots+x_n\ln\alpha_n=\ln\beta, \qquad x_1,\dots,x_n\in\mathbb{Z}, $$
for fixed $\alpha_1,\dots,\alpha_n$ in an algebraic number field $\mathbb{K}$ and for variable $\beta\in\mathbb{K}$ equal either to 1 or a root of unity.