Title: Borel Complexity of the set of vectors normal for a fixed recurrence sequence Show Chinese title

Title: 固定递归序列的正常向量集的Borel复杂性

Abstract: In this paper, we consider recurrence sequences $x_n=\xi_1 \alpha_1^n+\xi_2 \alpha_2^n$ ($n=0,1,\ldots$) with companion polynomial $P(X)$. For example, the sequence $x_n=\xi_1(4+\sqrt{2})^n+\xi_2(4-\sqrt{2})^n$ satisfies the recurrence $x_{n+2}-8x_{n+1}+14x_n=0$ and has companion polynomial $P(X)=X^2-8X+14=(X-4-\sqrt{2})(X-4+\sqrt{2})$. We call $(\xi_1,\xi_2)$ normal with respect to the recurrence relation determined by $P(X)$ when $(x_n)_{n\ge 0}$ is uniformly distributed modulo one. Determining the Borel complexity of the set of normal vectors for a fixed recurrence sequence is unresolved even for most geometric progressions. Under certain assumptions, we prove that the set of normal vectors is $\boldsymbol{\Pi}_3^0$-complete. A special case is the new result that the sets of numbers normal in base $\alpha$, i.e. $\{\xi\in \mathbb{R}\mid (\xi\alpha^n)_{n\geq 0}\mbox{ is u.d. modulo one.} \}$, are $\boldsymbol{\Pi}_3^0$-complete for every real number $\alpha$ with $|\alpha|$ Pisot. We analyze the fractional parts of recurrence sequences in terms of finite words via certain numeration systems. One of the difficulties in proving the main result is that even when recurrence sequences are uniformly distributed modulo one, it is not known what the average frequencies of the digits in the corresponding digital expansions are or if they even must exist. Show Chinese abstract

Abstract: 在本文中，我们考虑递推序列 $x_n=\xi_1 \alpha_1^n+\xi_2 \alpha_2^n$ ($n=0,1,\ldots$) 伴随多项式 $P(X)$。 例如，序列 $x_n=\xi_1(4+\sqrt{2})^n+\xi_2(4-\sqrt{2})^n$ 满足递推关系 $x_{n+2}-8x_{n+1}+14x_n=0$ 并且伴随多项式为 $P(X)=X^2-8X+14=(X-4-\sqrt{2})(X-4+\sqrt{2})$。 当$(x_n)_{n\ge 0}$在模一上均匀分布时，我们称$(\xi_1,\xi_2)$相对于由$P(X)$确定的递推关系是正常的。 确定固定递推序列的正常向量集的博雷尔复杂性，即使对于大多数几何级数仍未解决。 在某些假设下，我们证明正常向量集是$\boldsymbol{\Pi}_3^0$-完全的。 特殊情况是新的结果，即在基数$\alpha$中正常的数的集合，即 $\{\xi\in \mathbb{R}\mid (\xi\alpha^n)_{n\geq 0}\mbox{ is u.d. modulo one.} \}$，是$\boldsymbol{\Pi}_3^0$完全的，对于每个实数$\alpha$与$|\alpha|$为 Pisot。 我们通过某些数制系统，根据有限词来分析递推序列的小数部分。 证明主要结果的一个困难在于，即使递推序列在模一的情况下是均匀分布的，也不知道相应数字展开中数字的平均频率是什么，或者它们是否存在。