## 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.

### MSC:

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