A Cantor set type result in the field of formal Laurent series

TL;DR

This paper establishes a Khintchine type theorem for approximating elements in the Cantor set within formal Laurent series over , and constructs elements with any irrationality exponent  or higher.

Contribution

It introduces a new approximation theorem for the Cantor set in formal Laurent series and constructs elements with prescribed irrationality exponents.

Findings

Proves a Khintchine type theorem for the Cantor set in formal Laurent series.

Constructs elements with any irrationality exponent  or greater.

Provides new insights into Diophantine approximation in non-Archimedean settings.

Abstract

We prove a Khintchine type theorem for approximation of elements in the Cantor set, as a subset of the formal Laurent series over $\mathbb{F}_3$, by rational functions of a specific type. Furthermore we construct elements in the Cantor set with any prescribed irrationality exponent $\geq 2$.