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

Badly Approximable Numbers and the Growth Rate of the Inclusion Length of an Almost Periodic Function