Sum-product phenomena: p-adic case

TL;DR

This paper investigates sum-product phenomena in finite extensions of p-adic fields, emphasizing results with constants that do not depend on the prime p, thus generalizing previous work.

Contribution

It provides new sum-product estimates over p-adic fields with uniform constants across all primes p, extending the understanding of additive and multiplicative structures in these fields.

Findings

Sum-product estimates with p-independent constants

Generalization of sum-product phenomena to finite p-adic extensions

Enhanced understanding of additive and multiplicative interactions in p-adic fields

Abstract

The sum-product phenomena over a finite extension K of $\mathbb{Q}_p$ is explored. The main feature of the results is the fact that the implied constants are independent of $p$.