Entropy and independence in symbolic dynamics with connections to number theory

 We introduce a new family of shift spaces - the subordinate
shifts. Using them we prove in an elementary way that for every
nonnegative real number t one can find a shift space with entropy t.
Moreover, we show that there is a connection between positive entropy
and combinatorial independence of a shift space. Positive entropy can
be characterized through existence of a large (in terms of asymptotic
and Shnirelman densities) set of coordinates along which the highest
possible degree of randomness in points from the shift is observed. .
It turns out that the shift space known as "square-free flow" and its
relatives that have been recently extensively studied are all examples
of a subordinate shifts.

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