BOREL COMPLEXITY OF SETS OF POINTS WITH PRESCRIBED BIRKHOFF AVERAGES IN POLISH DYNAMICAL SYSTEMS WITH A SPECIFICATION PROPERTY

The paper’s main results under the strong approximate product structure (SAPS) are: (i) if |MT(X)| ≥ 2 then for every μ ∈ MT(X), G_μ is Π^0_3-complete; (ii) for any nonempty closed connected V ⊆ MT(X) with U = MT(X) \ V ≠ ∅, the sets G(V) and G_V are Π^0_3-hard (hence Π^0_3-complete in the compact case), U_G is Σ^0_3-hard, G^V is Π^0_2-complete, and all of these sets are dense. These are established via a quantitative tracing scheme (Lemmas 22–24 and 27) and a continuous reduction from a standard Π^0_3-complete set C3 ⊆ ω^ω, formalized in Theorem 29; see the statement and proof details in Section 7, including the construction around (19)–(21) and the blending schedule ψ_β, which ensures D(μ^β_j, μ^β_{j+1}) → 0, enabling Lemma 27 to apply . The candidate’s reductions and tracing lemmas essentially mirror these arguments (concatenation estimate ≈ Lemma 22/Corollary 23; “long block approximating μ” ≈ Lemma 24; “painting a sequence of measures” ≈ Lemma 27) and their hardness claims match Theorem 29. However, the candidate incorrectly asserts unconditional Borel upper bounds for G(V), G_V, and U_G for arbitrary Polish systems. The paper explicitly shows a compact/non-compact dichotomy: when X is not compact, G(V) and G_V can be Π^1_1 and UG can be Σ^1_1 (and even complete at that level), whereas only G^V remains Π^0_2 in full generality (Facts 16, 18, 19 and Theorem 33) . By contrast, the candidate claims G(V), G_V ∈ Π^0_3 and U_G ∈ Σ^0_3 for all Polish systems, which contradicts the paper’s results (and the explicit Π^1_1-complete example in Theorem 33) . Minor internal tension also appears where the candidate’s Lemma 3 first claims tracking across whole blocks but the proof only establishes convergence along block endpoints, whereas the paper’s Lemma 27 ensures uniform tracking via a slowly varying sequence μ_n(j). The core lower-bound constructions and density statements otherwise agree with the paper, but the candidate’s unconditional upper-bound claims are false in the non-compact setting.