| Popis: |
Let A be a finite alphabet, σ a substitution over A, (un) n∈N a fixed point for σ, and for each a∈ A, f(a) a real number; $$ {s^f}\left( n \right)\sum\limits_{n\, \leqslant N} {F\left( {{u_i}} \right)} $$ . We establish, specially in the “marginal case” (second eigenvalue of the matrix of the substitution is one) and under some additional assumptions, asymptotic formulae for $$ \sum\limits_{n{\kern 1pt} < N} {{s^f}\left( n \right)} ,{\mkern 1mu} \sum\limits_{n{\kern 1pt} < N} {{{\left( {{s^f}\left( n \right) - \gamma {{\log }_\theta }n} \right)}^2}} {\mkern 1mu} and{\mkern 1mu} \sum\limits_{n{\kern 1pt} < N} {{s^f}{{\left( n \right)}^k}.} $$ The results give answers to problems that appear in some works of C. Godreche, J.M. Luck and F. Vallet. The calculi rely upon systems of representation of integers. |
