TITLE: How many points must the arc intersect?
QUESTION [5 upvotes]: Let's say we have some number of points $\{x_i\}$, which lie on a circle.
We wish to position an arc somewhere on the circle, which has an arc length equal to $\frac1c$ of the circle's circumference, such that we minimise the number of the $\{x_i\}$ that lie on the arc.
Can we always position the arc such that at most $\frac1c$ of the points lie on it?
Note: when $c$ is an integer, we can do this by the pigeonhole principle: we cover the circle in $c$ of the arcs, and so one of them must have $\frac1c$ or fewer of the $\{x_i\}$ lying on it.

REPLY [4 votes]: If we have that $\frac1c=\frac{a}{b}$, then we can consider a generalisation of your pigeonhole principle idea. If we take $b$ copies of the arc we can cover the circle $a$ times- you could also think of this as having a circle with $a$ times the circumference, and the points $\{x_i\}$ are repeated periodically along it.
Then, there are effectively $a|\{x_i\}|$ points distributed amongst $b$ arcs, and by your "pigeonhole" idea, we get that there must be an arc with $\frac{a}{b}$ or fewer points on it.
We can then extend to the real case by a continuity argument. Note that you can bound the proportion arbitrarily closely by rationals on either side.