Rate Distortion Dimension and Ergodic Decomposition for R^d-Actions

The paper proves a mixture formula for the rate–distortion function for R^d-actions (Theorem 3.9) and derives the convexity/concavity statements for the upper/lower rate–distortion dimensions (Theorems 3.11 and 3.12) from it. The candidate solution invokes precisely this mixture formula and then reproduces essentially the same arguments: for the upper dimension, it uses domination by the metric mean dimension to justify a reverse-Fatou step; for the lower dimension, it uses a near-minimizer, Markov + Borel–Cantelli, and Fatou’s lemma along a rapidly decaying ε-sequence. Minor slips in the candidate write-up (choice of reproduction signal for the covering bound; a too-strong use of the liminf definition; and a sign direction in a ‘near-optimality’ inequality that should be justified via the “infimum” form) are easily corrected and do not affect the conclusion. The paper’s mixture formula and the subsequent proofs are complete and correct, and the model’s solution aligns closely with them. Key references in the paper: the mixture formula (Theorem 3.9) and its use to obtain Theorem 1.1/3.11 and Theorem 1.2/3.12 (, ); the uniform covering bound R ≤ S used for domination ().