# A dynamical construction of small totally $p$-adic algebraic numbers

## TL;DR

This paper presents a dynamical method to construct an infinite sequence of totally p-adic algebraic numbers with Weil heights approaching a specific limit, providing a new proof of a known result using explicit Arakelov-Zhang pairing calculations.

## Contribution

It introduces a novel dynamical construction of totally p-adic algebraic numbers and offers an alternative proof of a height limit result by explicitly computing the Arakelov-Zhang pairing.

## Key findings

- Constructed an infinite sequence of totally p-adic algebraic numbers
- Established Weil heights tend to (log p)/(p-1)
- Provided a new proof of Bombieri-Zannier's result

## Abstract

We give a dynamical construction of an infinite sequence of distinct totally $p$-adic algebraic numbers whose Weil heights tend to the limit $\frac{\log p}{p-1}$, thus giving a new proof of a result of Bombieri-Zannier. The proof is essentially equivalent to the explicit calculation of the Arakelov-Zhang pairing of the maps $\sigma(x)=x^2$ and $\phi_p(x)=\frac{1}{p}(x^p-x)$.