MINIMAL SUBSYSTEMS OF GIVEN MEAN DIMENSION IN BERNSTEIN SPACES

The paper proves (constructively) that for any compact frequency interval I, the shift on the Bernstein spaces BC(I) and B(I) contains minimal subsystems of every mean dimension strictly below twice the bandwidth, and also provides a system embedding simultaneously into B([−a/2,a/2]) and the full shift. Its construction via a carefully engineered symbolic bridge Y, an interpolation kernel with prescribed zeros, and explicit equivariant embeddings F and G is correct and complete. By contrast, the candidate solution relies on an asserted generalization of Dou’s result to produce minimal subshifts in ([0,1]^m)^Z with arbitrary mean dimension r<m (Dou’s published result covers r<1 in [0,1]^Z), and then sketches a “sum of bumps” band-limited coding. This approach omits key constraints (e.g., sampling/aliasing limits tied to bandwidth), contains a sign error in equivariance, and does not justify the crucial separation of “free windows” at positive density needed for injectivity at high dimensions. As written, the model’s proof is not rigorous and rests on unsupported assumptions.