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Summation formulae for substitutions on a finite alphabet. (English) Zbl 0718.11009
Number theory and physics, Proc. Winter Sch., Les Houches/Fr. 1989, Springer Proc. Phys. 47, 185-194 (1990).
[For the entire collection see Zbl 0702.00008.]
The author studies the sums \(s^ f(n)=\sum_{i\leq n}f(u_ i)\) where \((u_ i)\) is a fixed point of a substitution (see the previous review Zbl 0718.11008). In this paper he concentrates on the behavior of \(\sum_{n<N}s^ f(n)\). In particular, he proves that the centered moments of the distribution of \(s^ f(n)\) have the same property as the centered moment of a centered Gaussian distribution (assuming that the second eigenvalue of the matrix of the (primitive) substitution is equal to 1, and some additional reasonable conditions). - The results of the author answer a conjecture of C. Godrèche, J.-M. Luck and F. Vallet [Quasiperiodicity and types of order: a study in one dimension, J. Phys. A 20, 4483–4499 (1987)] and a conjecture of J.-M. Luck.

MSC:
11B85 Automata sequences
68Q65 Abstract data types; algebraic specification
26A16 Lipschitz (Hölder) classes
60E05 Probability distributions: general theory