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.

We fixed some inaccuracies and simplified some arguments in the previous version