The minimum number of distinct abelian squares in a binary circular word
We prove that every binary circular word of length $n$ contains at least $\fl{(n-1)/2}$ distinct abelian squares, and we determine all words attaining this bound. This settles a conjecture of Simpson: its odd case holds, and in its even case the list of extremal words misses $(ab)^{n/2}$ for $n\equiv 2\pmod 4$, $n\ge 6$.
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