On a problem of Schoenberg and Wills in diophantine approximation (Q1307423)

For a real number \(x\) denote by \(\|x\|\) the distance of \(x\) to the nearest integer. The main result of the paper is the following. Let \(n\leq 5\) and \(\alpha_1,\dots,\alpha_n\) be irrational and \(\beta_1,\dots,\beta_n\) be rational numbers which satisfy the inequality \[ \max_{1\leq i\leq n} \|q\alpha_i -\beta_i\|>\frac 12-\frac 1{2n} \] for all integer \(q\). Then \(\|\alpha_i\|=\|\alpha_j\|\) for \(i\neq j\). This result answers a question of J. M. Wills.