``Integer-making'' theorems (Q1158443)

Let \(f(n)=\max |\sum_{i\in E} (a_i-\alpha_i)|\) with \(a_i\) integer, \(\alpha_i\) real and \(| a_i-\alpha_i|<1\) for all \(i\), where the maximum is taken over all finite sets \(E\) having maximum degree \(n\). The authors prove that \(f(n)<n-1\) for \(n\geq 2\) (and conjecture that \(f(n)<c\sqrt n\)). A similar result (and also a similar conjecture) is given for another such function.