AI 中文总结
研究数域中无\(\mathfrak{B}\)整数的内在遍历性，证明相关子转移是内在遍历的，给出唯一最大熵测度。这是一维以上此类系统首个证明，依赖他人工作，还解决了相关情况，且给出基础刚性陈述的两个独立证明。
AI 中文摘要
设\(K\)是一个数域，其整数环为\(\mathscr{O}_K\)，\(\mathfrak{B}\)是\(\mathscr{O}_K\)中的一个厄尔多斯理想族。我们证明了相关的无\(\mathfrak{B}\)子转移\((X_{\mathfrak{B}},(S_a)_{a\in\mathscr{O}_K})\)是内在遍历的：它承载着唯一的最大熵测度，我们明确地将其识别为\(\prod_{\mathfrak{b}\in\mathfrak{B}}\mathscr{O}_K/\mathfrak{b}\)上哈尔旋转的相对独立扩展。这是一维以上无\(\mathfrak{B}\)系统内在遍历性的首个证明，依赖于阿劳霍 - 戴梅克 - 库拉加 - 普日姆斯的工作。通过他们的约化，我们还解决了\(k\)自由和无\(\mathfrak{B}\)格点情况以及\(k\)自由数域情况。我们给出了基础刚性陈述的两个独立证明：一个通过单位点相对熵论证，另一个通过佩克纳诱导 - 分裂方案的精确平铺实现。
英文摘要
Let $K$ be a number field with ring of integers $\mathscr{O}_K$, and let $\mathfrak{B}$ be an Erdős family of ideals in $\mathscr{O}_K$. We prove that the associated $\mathfrak{B}$-free subshift $(X_{\mathfrak{B}},(S_a)_{a\in\mathscr{O}_K})$ is intrinsically ergodic: it carries a unique measure of maximal entropy, which we identify explicitly as a relatively independent extension of the Haar rotation on $\prod_{\mathfrak{b}\in\mathfrak{B}}\mathscr{O}_K/\mathfrak{b}$. This is the first proof of intrinsic ergodicity for $\mathfrak{B}$-free systems beyond dimension one, and relies on the work of Araújo--Dymek--Kułaga-Przymus. Via their reductions, we also settle the $k$-free and $\mathfrak{B}$-free lattice-point cases and the $k$-free number-field case. We give two independent proofs of the underlying rigidity statement: one through a single-site relative-entropy argument, and one through an exact-tiling realisation of Peckner's induce-and-split scheme.
Comments14 pages
