交替基系统中的周期性与纯周期性

纯数域整数基的最小周期性

The minimal periodicity for integral bases of pure number fields

本文研究了纯数域中整数基的最小周期性，通过局部到全局的原则，证明了全局形状的周期模为 $M(n)=n\cdot\mathrm{rad}(n)$，并给出了应用。

交替进制下周期康托尔实数展开的Solomyak定理类似物

An analogue of Solomyak's theorem for periodic Cantor real expansions in alternate bases

本文研究交替进制下单位区间上的康托尔实数展开，针对γ+1展开为周期的情形，计算了进制所对应的非平凡伽罗瓦共轭的范数界，从而将Solomyak关于经典beta展开的结果推广到交替进制框架。

利用循环节矩阵形式研究一般多维连分数的周期性

Periodicity of general multidimensional continued fractions using repetend matrix form

本文研究一般多维连分数展开的周期性，通过刻画循环节对应矩阵的结构，证明若干向量在 Algebraic Jacobi–Perron 算法中具有最终周期展开，并给出纯周期展开的判据。

On periodicity of $p$-adic Browkin continued fractions

The classical theory of continued fractions has been widely studied for centuries for its important properties of good approximation, and more recently it has been generalized to $p$-adic numbers where it presents many differences with respect to the real case. In this paper we investigate periodicity for the $p$-adic continued fractions introduced by Browkin. We give some necessary and sufficient conditions for periodicity in general, although a full characterization of $p$-adic numbers having purely periodic Browkin continued fraction expansion is still missing. In the second part of the pap…

Purely Periodic and Transcendental Complex Continued Fractions

Adolf Hurwitz proposed in 1887 a continued fraction algorithm for complex numbers: Hurwitz continued fractions (HCF). Among other similarities between HCF and regular continued fractions, quadratic irrational numbers over $\mathbb{Q}(i)$ are precisely those with periodic HCF expansions. In this paper, we give some necessary as well as some sufficient conditions for pure periodicity of HCF. Then, we characterize badly approximable complex numbers in terms of HCF. Finally, we prove a slightly weaker complex analogue of a theorem by Y. Bugeaud on the transcendence of certain continued fractions.

负基系统中的周期展开式

Purely periodic expansions in systems with negative base

研究负基数制中展开式的纯周期性问题。与Akiyama关于正Pisot单位基$β$的结果类似，我们找到了一个充分条件，使得存在一个包含原点的区间$J$，使得从$J$开始的每个有理数的$(-β)$ -展开式是纯周期的.我们专注于二次基的情况下，并证明了以下的差异之间的负和正的基础：它是已知的有限性性质（${\rm Fin}(β)=\Z[β]$）不仅是充分的，但也是必要的情况下，积极的二次和三次基地。我们表明，${\rm Fin}(-β)=\Z[β]$是不必要的情况下，负基地。