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### Abstract

Consider the problem of calculating the fractal dimension of a set X consisting
of all infinite sequences S over a finite alphabet \Sigma that satisfy some
given
condition P on the asymptotic frequencies with which various symbols from \Sigma
appear in S. Solutions to this problem are known in cases where (i) the fractal
dimension is classical (Hausdorff or packing dimension), or (ii) the fractal
dimension is effective (even finite-state) and the condition P completely
specifies an empirical distribution \pi over \Sigma, i.e., a limiting frequency
of occurrence for every symbol in \Sigma.
In this paper we show how to calculate the finite-state dimension (equivalently,
the finite-state compressibility) of such a set X when the condition P only
imposes partial constraints on the limiting frequencies of symbols. Our results
automatically extend to less restrictive effective fractal dimensions (e.g.,
polynomial-time, computable, and constructive dimensions), and they have the
classical results (i) as immediate corollaries. Our methods are nevertheless
elementary and, in most cases, simpler than those by which the classical results
were obtained.