The Zariski closure of integral points on varieties parametrizing periodic continued fractions

TL;DR

This paper investigates the distribution of integral points on varieties parametrizing periodic continued fractions, revealing conditions under which these points are Zariski dense or not, depending on the nature of the units in associated rings.

Contribution

It establishes new criteria for Zariski density of integral points on these varieties based on the properties of units in related rings and proves rationality and irreducibility for large parameters.

Findings

Integral points are not Zariski dense over integers or imaginary quadratic rings.

Zariski density occurs under certain unit and norm conditions for large parameters.

Varieties become rational and irreducible for sufficiently large k.

Abstract

Let $R$ be the ring of $S$-integers in a number field $K$. Let $\mathcal{B}=\{\beta, \beta^{\ast}\}$ be the multi-set of roots of a nonzero quadratic polynomial over $R$. There are varieties $V(\mathcal{B})_{N,k}$ defined over $R$ parametrizing periodic continued fractions $[b_1,\ldots , b_N,\overline{a_1,\ldots ,a_k}]$ for $\beta$ or $\beta^{\ast}$. We study the $R$-points on these varieties, finding contrasting behavior according to whether groups of units are infinite or not. If $R$ is the rational integers or the ring of integers in an imaginary quadratic field, we prove that the $R$-points of $V(\mathcal{B})_{N,k}$ are not Zariski dense. On the other hand, suppose that $\beta\not\in K\cup\{\infty\}$, $R^\times$ is infinite, and that there are infinitely many units in the (left) order $R_\beta$ of $\beta R+R\subseteq K(\beta)$ with norm to $K$ equal to $(-1)^k$. Then we prove that the $R$-points on $V(\mathcal{B})_{1,k}$ are Zariski dense for $k\geq 8$ and the $R$-points on $V(\mathcal{B})_{0,k}$ are Zariski dense for $k\geq 9$. We also prove that $V(\mathcal{B})_{1,k}$ and $V(\mathcal{B})_{0,k}$ are $K$-rational irreducible varieties for $k$ sufficiently large.