Submission date: 27 April 2020.

Abstract:

Let K be an elementary extension of ℚ_p, V be the set of finite a in K, st be the standard part map K^m → ℚ^m_p, and X ⊆ K^m be K-definable. Delon has shown that ℚ^m_p ∩ X is ℚ_p-definable. Yao has shown that dim ℚ^m_p ∩ X ≤ dim X and dim st(V^n ∩ X) ≤ dim X. We give new NIP-theoretic proofs of these results and show that both inequalities hold in much more general settings. We also prove the analogous results for the expansion ℚ_p^{an} of ℚ_p by all analytic functions ℤ_p^n → ℚ_p. As an application we show that if (X_k)_{k in ℕ} is a sequence of elements of an ℚ_p^{an}-definable family of subsets of ℚ_p^m which converges in the Hausdorff topology to X ⊆ ℚ_p^m then X is ℚ_p^{an}-definable and dim X ≤ limsup_{k \to ∞} dim X_k.

Mathematics Subject Classification:

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Full text arXiv 2004.13109: pdf, ps.