IRREGULAR SET AND METRIC MEAN DIMENSION WITH POTENTIAL

The paper proves that, for systems with the specification property, the irregular set either is empty or carries full Bowen upper and lower metric mean dimension with potential, equal to the classical metric mean dimension on X (Theorem 1.1) . It does so via a Moran-like construction and a generalized pressure distribution principle at fixed scale, together with variational principles for the metric mean dimension with potential (Section 2; Proposition 3.18; concluding inequalities) . The candidate solution gives a shorter argument: (i) identify the scale-ε Carathéodory/Bowen pressure Mε with the Pesin–Pitskel construction; (ii) invoke Thompson’s full-pressure theorem for irregular sets for each fixed potential Φε = (log(1/ε))ψ; (iii) relate the pressure on X at scale ε to limsup spanning-sum growth; and (iv) divide by log(1/ε) and send ε→0. The approaches are different. The model’s logic is essentially sound but slightly overclaims in Step 2 by asserting full-pressure equality at every fixed ε directly from Thompson; Thompson yields full Pesin–Pitskel pressure (the ε→0 limit) for each fixed potential, from which the desired conclusion still follows because the scale-ε outer pressure differs from the limit by a bounded amount that vanishes after normalization by log(1/ε). With this mild correction, the model’s proof aligns with the paper’s result.