论文标题
代数基础中的字母，重写步道和定期表示
Alphabets, rewriting trails and periodic representations in algebraic bases
论文作者
论文摘要
对于$β> 1 $ a真正的代数整数（{\ it base}），有限字母$ \ Mathcal {a} \ subset \ subset \ mathbb {z} $实现了身份$ \ mathbb {q}（q}（q}（q}（β）（β）（β）（β）（β）= {\ rm per} $ math） $ {\ rm per} _ {\ Mathcal {a}}（β）$是$（β，\ m rathcal {a}）$的复数集，最终研究了定期表示。与贪婪算法相比，定义了最小和最大字母。显示最大字母与基础$β$和莱默的问题的渐近数相关。引入了重写跟踪的概念，以构建与基座的小多项式相关的中间字母。研究了对$ \ mathbb {q}（β）$中原点社区表示的后果，研究了施密特定理与PISOT数字相关的定理。给出了Galois共轭的应用，用于基础$γ_s的收敛序列：=γ_{n，m_1，\ ldots，m_s} $，以至于$γ_{s}^{ - 1} $是$（0,1）的唯一根（0,1）$（0,1）$ Newman polynomial in Type $ $（0,1）。 +x^{m_1}+\ ldots+x^{m_s} $，$ n \ geq 3 $，$ s \ geq 1 $，$ m_1- n \ n \ geq n-1 $，$ m_ {q+1} -mm_q \ geq n-1 $ for All $ q \ geq for ash $ q \ geq 1 $。对于$β> 1 $ a的相互代数整数接近一个，模量$ <1 $ <1 $ <1 $的动力学Zeta功能的$β$ -Shift $ζ_β（z）$在某些假设下显示为最小的$β$的最小值多态度的零。
For $β> 1$ a real algebraic integer ({\it the base}), the finite alphabets $\mathcal{A} \subset \mathbb{Z}$ which realize the identity $\mathbb{Q}(β) = {\rm Per}_{\mathcal{A}}(β)$, where ${\rm Per}_{\mathcal{A}}(β)$ is the set of complex numbers which are $(β, \mathcal{A})$-eventually periodic representations, are investigated. Comparing with the greedy algorithm, minimal and maximal alphabets are defined. The maximal alphabets are shown to be correlated to the asymptotics of the Pierce numbers of the base $β$ and Lehmer's problem. The notion of rewriting trail is introduced to construct intermediate alphabets associated with small polynomial values of the base. Consequences on the representations of neighbourhoods of the origin in $\mathbb{Q}(β)$, generalizing Schmidt's theorem related to Pisot numbers, are investigated. Applications to Galois conjugation are given for convergent sequences of bases $γ_s := γ_{n, m_1 , \ldots , m_s}$ such that $γ_{s}^{-1}$ is the unique root in $(0,1)$ of an almost Newman polynomial of the type $-1+x+x^n +x^{m_1}+\ldots+ x^{m_s}$, $n \geq 3$, $s \geq 1$, $m_1 - n \geq n-1$, $m_{q+1}-m_q \geq n-1$ for all $q \geq 1$. For $β> 1$ a reciprocal algebraic integer close to one, the poles of modulus $< 1$ of the dynamical zeta function of the $β$-shift $ζ_β(z)$ are shown, under some assumptions, to be zeroes of the minimal polynomial of $β$.
