Abstract
A word w over the alphabet A is called uniform if for any two words u and v of the same length, the numbers of occurrences of u and v in w differ at most by 1. In particular, a uniform word contains as factors all the words of length ⩽ G w , where G w is the maximal length of a repeated factor of w. Some characterizations of uniform words are given. A lower bound for the number of uniform words of length N is determined in some special cases. The main result of the paper is the proof that on each alphabet A there exist uniform words of any length. Moreover, an efficient algorithm to construct for any N a uniform word of length N is given. Finally, we give a characterization of a uniform word of length N as a minimum of two different quasi-order relations defined in A N and as a maximum of suitable entropy functionals.
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