# Multiplicative $p$-adic metric Diophantine approximation on manifolds and dichotomy of exponents

## TL;DR

This paper investigates $p$-adic Diophantine approximation on manifolds, focusing on multiplicative approximation on affine subspaces and establishing a dichotomy for analytic $p$-adic manifolds, advancing understanding in this specialized area.

## Contribution

It introduces a new framework for multiplicative $p$-adic Diophantine approximation on manifolds and proves a dichotomy result for analytic $p$-adic manifolds.

## Key findings

- Established a dichotomy for $p$-adic Diophantine exponents on manifolds
- Extended multiplicative approximation results to affine subspaces
- Provided new insights into $p$-adic Diophantine approximation behavior

## Abstract

In this paper we study $p$-adic Diophantine approximation on manifolds, specifically multiplicative Diophantine approximation on affine subspaces and a Diophantine dichotomy for analytic $p$-adic manifolds.