Sum-product phenomenon in quotients of rings of algebraic integers

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.