Sum-product phenomenon in quotients of rings of algebraic integers

TL;DR

This paper proves a bounded generation theorem for quotients of rings of algebraic integers, addressing a conjecture and providing bounds for additive character sums using sum-product estimates.

Contribution

It establishes a bounded generation result over quotient rings of algebraic integers and derives new bounds for additive character sums, advancing understanding in algebraic number theory.

Findings

Proved a bounded generation theorem over $\

Obtained nontrivial bounds for additive character sums over $\

Abstract

We obtain a bounded generation theorem over $\mathcal O/\mathfrak a$, where $\mathcal O$ is the ring of integers of a number field and $\mathfrak a$ a general ideal of $\mathcal O$. This addresses a conjecture of Salehi-Golsefidy. Along the way, we obtain nontrivial bounds for additive character sums over $\mathcal O/\mathcal P^n$ for a prime ideal $\mathcal P$ with the aid of certain sum-product estimates.