标题： 高斯整数的科布姆定理 显示英文标题

标题： Cobham's theorem for the Gaussian integers

摘要： 假设四指数猜想成立，Hansel 和 Safer 表明，如果高斯整数的一个子集$S$同时是$\alpha=-m+i $-和$\beta=-n+i$-可识别的，那么它是稠密的，并且他们猜想$S$必须是最终周期性的。 在不假设四指数猜想的情况下，我们证明如果$\alpha$和$\beta$是乘法独立的高斯整数，并且至少有一个$\alpha$,$\beta$不是一个$n$-次整数根，那么任何$\alpha$-和$\beta$-自动配置最终是周期性的；特别是我们证明了Hansel和Safer的猜想。 Otherwise, there exist non-eventually periodic configurations which are $\alpha$-automatic for any root of an integer $\alpha$. Our work generalises the Cobham-Semenov theorem to Gaussian numerations. 显示英文摘要

摘要： Assuming the four exponentials conjecture, Hansel and Safer showed that if a subset $S$ of the Gaussian integers is both $\alpha=-m+i $- and $\beta=-n+i$-recognizable, then it is syndetic, and they conjectured that $S$ must be eventually periodic. Without assuming the four exponentials conjecture, we show that if $\alpha$ and $\beta$ are multiplicatively independent Gaussian integers, and at least one of $\alpha$, $\beta$ is not an $n$-th root of an integer, then any $\alpha$- and $\beta$-automatic configuration is eventually periodic; in particular we prove Hansel and Safer's conjecture. Otherwise, there exist non-eventually periodic configurations which are $\alpha$-automatic for any root of an integer $\alpha$. Our work generalises the Cobham-Semenov theorem to Gaussian numerations.