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arXiv · 2308.08867

Sum-product phenomenon in quotients of rings of algebraic integers

Jincheng Tang·Xin Zhang
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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.

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Jincheng Tang, Xin Zhang. 2023-08-17. Sum-product phenomenon in quotients of rings of algebraic integers. https://arxiv.org/abs/2308.08867

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