To the Blichfeldt–Mullender–Spohn Theorem on Simultaneous Approximation

A new approach to strengthening a result of Spohn based on the analysis of best approximations is suggested. Let $\alpha _1,\dots ,\alpha _m$ be real numbers. Let $c_m$ denote the least upper bound of all constants $\sigma $ for which the inequality $\max _{j=1,\dots ,m}\|p\alpha _j\| < (\sigma p)^{-1/m}$ has infinitely many positive integer solutions $p$; here, $\|\cdot \|$ is the distance to the nearest integer. Lower bounds for $c_m$ that hold for all $m$ are studied.