  We study the growth rate of the inclusion length of an almost periodic
function. For a given a. p. function such growth rate depends on the algebraic
structure of Fourier exponents, i. e. on how good they can be approximated by
rational numbers. In additional, as appears from the definition, the inclusion
length carries some information about the translation numbers (almost periods).
Our result is a lower bound of the growth rate of the inclusion interval of a
quasiperiodic function (theorem 3). Here we use methods from dimension theory.
We do not assume anything about exponents, but rationally independence. This
suggest an idea that this lower bound can be reached (in asymptotic sense) for
some "bad" exponents. Koichiro Naito in his papers on estimates of the fractal
dimension of almost periodic attractors proved an upper bound of the inclusion
length for some class of a.p. functions, using simultaneous Diophantine
approximations. For the special case of badly approximable exponents we can see
that the both estimates (if we consider them as asymptotic estimates) are
coincide (see theorem 4). We hope that ideas and results presented in this
paper can be useful not only to understand the nature of badly approximable
numbers and almost periods, but also for more detailed understanding of the
structure of almost periodic attractors.
