Good Functions, Measures, and the Kleinbock–Tomanov Conjecture

The paper’s main theorem (Theorem 2.1) states exactly the “solver question”: if μ on U⊂Q_ν^d is absolutely decaying and Federer, and f:U→Q_ν^n is C^{l+1}, nonsingular and l-nondegenerate at μ-a.e. point, then f_*μ is friendly. The authors reduce the proof to two ingredients: (i) local bi-Lipschitz charts (coming from nonsingularity) and (ii) a nontrivial p-adic ‘good functions’ statement, packaged as Proposition 4.1, which they prove via new ultrametric techniques precisely because the classical mean value theorem used over R in KLW does not carry over to Q_p. They then invoke Lemma 4.1 to conclude friendliness of f_*μ from (i)–(ii) . The candidate solution sketches the same high-level plan (local bi-Lipschitz + goodness for g=ℓ∘f), but its crucial Step 3 claims that the KLW ‘good functions’ mechanism “carries over verbatim” to the ultrametric case. The paper explicitly identifies the lack of a mean value theorem as a barrier and develops a different proof to overcome it, culminating in Proposition 4.1 (and Theorem 5.1), rather than a verbatim carryover from KLW . Moreover, the model asserts a convenient exponent α_g=α/(dl) without justification, while the paper proves a specific quantitative exponent α′=α/(2^{l+1}−2) via a delicate argument (see Theorem 5.1 and its use in Proposition 4.1) . Thus, the paper’s result and proof are correct, whereas the model’s proof glosses over the technical heart and makes an incorrect “verbatim” transfer claim and an unsupported exponent choice.