Do balanced words have a short period?
Résumé
We conjecture that each balanced word on $N$ letters
- either arises from a balanced word on two letters by expanding both letters with a congruence word,
- or is $D$-periodic with $D \leq 2^N-1$.
Our conjecture arises from extensive numerical experiments. It implies, for any fixed $N$, the finiteness of the number of balanced words on $N$ letters which do not arise from expanding a balanced word on two letters. It refines a theorem of Graham and Hubert, which states that non-periodic balanced words are congruence expansions of balanced words on two letters. It also relates to Fraenkel's conjecture, which states that for $N \geq 3$, every balanced word with distinct densities $d_1> d_2 \dots >d_N$ satisfies $d_i = (2^{N-i}) / (2^N - 1)$, since this implies that the word is $D$-periodic with $D= 2^N -1$. For $N\leq 6$, we provide a tentative list of the density vectors of balanced words which do not arise from expanding a balanced word with fewer letters. We prove that the list is complete for $N=4$ letters.
We also prove that deleting a letter in a congruence word always produces a balanced word and this construction allows us to further reduce the list of density vectors that remains unexplained. Moreover, we prove that deleting a letter in a $m$-balanced word produces a $m+1$-balanced word, thus extending and simplifying a result of Sano et al.
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