ZARISKI DENSITY OF POINTS WITH MAXIMAL ARITHMETIC DEGREE

The paper’s Question 1.5 asks whether (X,f) has densely many K-rational points with maximal arithmetic degree (disjoint orbits included), and Theorem 1.8 answers this affirmatively for every surjective self-morphism with δf>1 over a number field K. But this cannot hold in general: take an elliptic curve E/K with E(K) finite and f=[m] with m≥2; then δf=m^2>1 yet E(K) contains no Zariski-dense subset at all, contradicting Theorem 1.8. The proof of Section 3 in the paper appears to rely on constructing points of arbitrarily large extension degree (via a set T(D,A,B) defined in Lemma 3.3) and thus actually builds algebraic points over K̄, not K-rational points; Lemma 3.3 itself is stated with x∈X(K) but then uses [K(x):K]≫1, which is impossible for K-rational points and signals a notation/quantifier error in that section. Meanwhile, the paper’s other results (unirational: Theorem 1.11; abelian: Theorem 1.12; P1-bundles over elliptic curves: Theorem 6.1) align with the model’s weaker but correct statements about density after passing to a suitable finite extension. Thus the model’s counterexample and corrected scope are right, while the paper’s main Theorem 1.8 (as written) is false. Key loci: Definition 1.4 and Question 1.5 define the property and ask it over K; Theorem 1.8 claims a positive answer over K; Lemma 3.3 and the surrounding argument in Section 3 implicitly use algebraic points of large degree, not K-points. See the paper’s Definition 1.4, Question 1.5, and Theorem 1.8; and the density/proof devices in Lemma 3.1 and Lemma 3.3 (and Theorems 2.1–2.2 for canonical heights) for the mismatch between the statement and what is actually proved. The unirational/abelian cases (Theorems 1.11 and 1.12) and the P1-bundle case (Theorem 6.1) are consistent with the model’s summary. Citations: Definition 1.4 and Question 1.5, Theorem 1.8, Theorems 2.1–2.2, Lemmas 3.1 and 3.3, Theorems 1.11 and 1.12, Theorem 6.1, Abstract.