On the interval $I = [0,1)$ define the distance $d(a,b) = min(|a-b|, 1-|a-b|)$. This metric is equivalent to the usual topology on $\Bbb{R}/\Bbb{Z}$ or $S^1$.
Given an array $A = [x_0,...,x_{n-1}]$ of elements of $I$, we define its spacing as $$s(A) = n \cdot \min_{i \neq j} \left[d(x_i, x_j) \cdot d\left(\frac in, \frac jn\right)\right]$$
Note that $s([x_0,x_1]) = d(x_0,x_1)$
- For a fixed $n$, how do we find array of length $n$ with maximum spacing ?
- Denoting the above maximum spacing as $c_n$, how does the sequence $\{c_n\}$ behaves for large $n$ ?
Its easy to see that $c_n \geq {1 \over n}$ (from equally spaced grid), but this bound is not tight. Using empirical examples, it can be shown that $c_8 \geq {4 \over 15}$, $c_{10} \geq {1 \over 5}$.
Clearly $c_n$ is monotonically decreasing and bounded below by zero, so $\lim_{n} c_n $ must exist. Is this limit zero ?