On the diophantine approximation of linear forms (Q769890)

The author improves a classical result of Dirichlet by proving the following theorem. Let \(p\), \(n_i\), \(m_i\) \((1\leq i\leq p)\) be natural numbers, and \(f = \sum_{\nu=1}^p \sum_{\mu=1}^{n_\nu} a_{\nu\mu}x_{\mu}^{(\nu)}\) a linear form with real coefficients \(a_{\nu\mu}\), then integer values \(x_{\mu}^{(\nu)}\), \(y\) will exist, not all of them equal to zero, such that: \[ | f-y|\leq M^{-1},\quad M=\prod_{\nu=1}^p (n_\nu m_\nu +1 ),\quad | x_{\mu}^{(\nu)}|\leq m_\nu,\quad x_i^{(\nu)}x_j^{(\nu)}\geq 0. \] The equality \(| f-y| = M^{-1}\) is actually attained for some forms for which \(| f-y| < M^{-1}\) never holds. Several consequences are deduced, in particular a well known Thue-Nagell theorem is generalized. The author has pointed out to the reviewer some misprints unfortunately overlooked in the proofs.