Entropy-minimal subshift characterization of uniform dimension in arithmetic progressions

Let $g\geq2$ be an integer, let $A\subseteq\mathbb{N}_0$ be $\times g$-invariant, meaning invariant under multiplication by $g$, and let $\dim_{\mathrm{M}}$ denote Minkowski dimension. An entropy minimal subshift is a subshift whose every proper subshift has strictly smaller topological entropy.

Entropy-minimal subshift conjecture. There exists an entropy minimal subshift representing $A$ if and only if, for every arithmetic progression $P$,

$$\dim_{\mathrm{M}}(A\cap P)\in\{0,\dim_{\mathrm{M}}(A)\}.$$

This conjecture proposes a characterization of the sets arising from entropy minimal symbolic systems through their intersections with arithmetic progressions. The paper establishes the analogous assertion for transitive sofic subshifts and gives broader affirmative results, but the stated equivalence is presented as a conjecture.