Aperiodicity Measure for Infinite Sequences

We introduce the notion of aperiodicity measure for infinite symbolic sequences. Informally speaking, the aperiodicity measure of a sequence is the maximum number (between 0 and 1) such that this sequence differs from each of its non-identical shifts in at least fraction of symbols being this number. We give lower and tipper bounds on the aperiodicity measure of a sequence over a fixed alphabet. We compute the aperiodicity measure for the Thue-Morse sequence and its natural generalization the Prouhet sequences, and also prove the aperiodicity measure of the Sturmian sequences to be 0. Finally. we construct an automatic sequence with the aperiodicity measure arbitrarily close to 1.