|
Let $n$ be a fixed natural number. {\sc Wills} has shown that there exist irrational numbers $\vect{\alpha}$ and real numbers $\vect{\beta}$ with $\max_{1\le i\le n} \|q\alpha_i-\beta_i\|>1/2-1/2n$ for all integers $q$ ($\|\cdot\|$ denotes the distance to the nearest integer). His example is $\alpha_i=\alpha$ and $\beta_i =i/n+\delta$, $\delta$ suitably chosen. Beyond that, he asked if $\alpha_i$ can be found with pairwise different $\|\alpha_i\|$. We prove that this does not hold for $n\le 5$, thereby revealing the close relation to Schoenberg's billiard ball problem for cubes and classifying its critical lines in these dimensions.
|
|||
 |
|||
Key words and phrases:
Simultaneous inhomogeneous diophantine approximation, billiard ball problems,
view-obstruction problems.