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We introduce and study a dimensional-like characteristic of an uniformlyalmost periodic function, which we call the Diophantine dimension. Bydefinition, it is the exponent in the asymptotic behavior of the inclusiolength. Diophantine dimension is connected with recurrent and ergodicproperties of an almost periodic function. We get some estimates of theDiophantine dimension for certain quasiperiodic functions and present methodsto investigate such a characteristic for almost periodic trajectories ofevolution equations. Also we discuss the link between the presented approachand the so called effective versions of the Kronecker theorem.

Dimension Theory Approach to the Complexity of Almost Periodic Trajectories