The convergence Khintchine theorem for polynomials and planar \(p\)-adic curves. (English) Zbl 0992.11043

The author discusses main steps of proofs of two theorems that have previously appeared in the Preprint Series of the Institute of Mathematics NAS Belarus, No. 8 (547) (1998) (Theorem 1) and No. 2 (556) (1998) (Theorem 2, jointly with V. Beresnevich). She called them convergence Khintchine theorems (in the real case it is also known as Borel-Cantelli lemma) and they have the following form: Let \(p\) be a prime number, \(\mathbb Q_p\) be the field of \(p\)-adic numbers with the \(p\)-adic valuation \((\cdot)_p\), \(\mathbb Z_p\) be the ring of \(p\)-adic integers and \(\Psi (h)\) be monotonic, \(h=1,2,\dots \).
Theorem 1: If \(\sum _{h=1}^\infty \Psi (h)<\infty \), then for almost all \(x\in \mathbb Q_p\) (with respect to Haar measure) the inequality \(|F(x)|_p<h^{-n}\Psi (h)\) holds only for finitely many polynomials \(F\in \mathbb Z[x]\), \(\text{deg }F=n\), where \(h\) is the height of \(F\).
Theorem 2: If \(\sum _{H=1}^\infty \Psi (H)<\infty \), then for almost all \(x\in \mathbb Z_p\) the inequality \(|G(x)|_p<H^{-2}\Psi (H)\) holds only for finitely many \(G(x)=b_0+b_1x+b_2f(x)\), where \(H=\max|b_i|\neq 0\), \(b_i\in \mathbb Z\) and \(f(x)=\sum _{n=1}^\infty c_n(x-c)^n\) is a normal function such that \(f''(x)\neq 0\) almost everywhere on \(\mathbb Z_p\).
The author also gives the history of such problems.


11J61 Approximation in non-Archimedean valuations
11J83 Metric theory
11K60 Diophantine approximation in probabilistic number theory