On Dirichlet's theorem (Q2470900)

The author considers a variant of Dirichlet's theorem. In particular he proves that for a linear form \(L\) with real coefficients there exist infinitely many \(q=(q_1,\ldots,q_n)\in \Omega\cup\mathbb Z^n\) such that the \(\|L(q)\|<(\max_i(|q_i|))^{-n}\), where \(\|\cdot\|\) denotes the distance to the nearest integer and \(\Omega\subset\mathbb R^n\) is of special form.