Stromquist and Woodall (1985) study the problem of Sets on which several measures agree. There are $n$ non-atomic value measures on the unit circle, and a parameter $w\in(0,1)$. The goal is to find a subset of the circle, whose value according to all $n$ measures is exactly a fraction $w$ of the total. The subset should be a union of a finite number of intervals, and this number should be as small as possible.
Denote by $Ints(n,w)$ the smallest number of required intervals. Stromquist and Woodall prove the following bounds on $Ints(n,w)$:
- $Ints(n,w) \leq n-1$. This is their main result, and it is proved using a nice topological argument which is summarized in Wikipedia.
- $Ints(n,w) \geq n/2$. Proof: suppose every measure is concentrated in a different location. Then any $w$-subset should have an endpoint in every location. $n$ endpoints require at least $n/2$ intervals.
- If $w$ is irrational, OR $w$ is a reduced fraction whose denominator is at least $n$ (in particular, if $w=1/n$) then $Ints(n,w) \geq n-1$ (i.e. the upper bound is tight). A sketch of the proof appears in that Wikipedia page for the case $w=1/n$
They leave open the case in which $w$ is a fraction whose denominator is less than $n$. The smallest such case is: $Ints(4,1/3)$. There, the lower bound is 2 and the upper bound is 3.
Has there been any advancement in this question since 1985?
In particular: is it possible to cut 2 intervals of a circle, such that all 4 measures value it as exactly $1/3$?
EDIT: I asked Prof. Stromquist and he said he is not aware of new results since 1985. So the question is still open.
