Borel Complexity of the set of vectors normal for a fixed recurrence sequence

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.