## Balanced words.(English)Zbl 1070.68129

A finite or infinite word defined over a finite alphabet $${\mathcal A}$$ is said to be balanced if, for any pair of factors $$(W,W')$$ with the same length and any element $$a$$ in $${\mathcal A}$$, we have $\left| | W|_a-| W'|_a\right|\leq 1.$ Here $$| W|_a$$ denotes the number of occurrences of the letter $$a$$ in the word $$W$$.
The paper under review presents an interesting survey of the notion of balance. Classical results are recalled, but the purpose of the author is mainly to focus on various generalizations of this notion with applications and with open problems in number theory, theoretical computer science or ergodic theory. Most of the mentioned results correspond to recent works.

### MSC:

 68R15 Combinatorics on words 05A05 Permutations, words, matrices 37B10 Symbolic dynamics

### Keywords:

combinatorics on words; balance property; Sturmian words
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