A subshift with linear block complexity has at most countably many ergodic measures, and we continue the study of the relation between such complexity and the invariant measures. By constructing minimal subshifts whose block complexity is arbitrarily close to linear but have uncountably many ergodic measures, we show that this behavior fails as soon as the block complexity is superlinear. With a different construction, we show that there exists a minimal subshift with an ergodic measure such that the liminf of the slow entropy grows slower than any given rate tending to infinitely but the limsup grows faster than any other rate majorizing this one yet still growing subexponentially. These constructions lead to obstructions in using subshifts in applications to properties of the prime numbers and in finding a measurable version of the complexity gap that arises for shifts of sublinear complexity.
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