On a Problem of Schoenberg and Wills in Diophantine Approximation

Let n be a fixed natural number. Wills has shown that there exist irrational numbers α₁,..., α_n and real numbers β₁,..., β₁ with max_1≤i≤n ‖qα_i-β_i‖ > 1/2 – 1/2n for all integers q (‖·‖ denotes the distance to the nearest integer). His example is α_i = α and β_i = i/n + δ, δ suitably chosen. Beyond that, he asked if α_i can be found with pairwise different ‖α_i‖. 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.