On a Problem of Schoenberg and Wills in Diophantine Approximation

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Let n be a fixed natural number. Wills has shown that there exist irrational numbers α1,..., αn and real numbers β1,..., β1 with max1≤i≤n ‖qαi-βi‖ > 1/2 – 1/2n for all integers q (‖·‖ denotes the distance to the nearest integer). His example is αi = α and βi = i/n + δ, δ suitably chosen. Beyond that, he asked if αi can be found with pairwise different ‖αi‖. We prove that this does not hold for n ≤ 5, thereby revealing the close relation to Schoenberg's billiard ball problem for cubes and classifying its critical lines in these dimensions.