论文标题
Dedekind域，完成和HAAR测量的密度
Densities on Dedekind domains, completions and Haar measure
论文作者
论文摘要
令$ d $为全球领域中的$ s $ inmintegers的戒指，其profinite完成。我们讨论了$ d $中的密度与$ \ hat {d} $的HAAR度量之间的关系：特别是，我们询问一个子集的$ x $ of $ d $的密度等于其在$ \ hat {D} $中的关闭量。为了具有确切的陈述，我们给出了涵盖最常用的密度的一般定义。使用它，我们为密度和度量之间的平等提供了必要和充分的条件，该密度和度量均包含了由于Poonen和Stoll引起的标准。在另一个方向上，我们将Davenport-erdő定理扩展到上述每个$ d $，并将其作为“密度=度量”结果提供新的解释。我们的观点还提供了一个简单的证据，即在任何$ d $中，最多可将$ k $ difins Primes的元素排除在任何天然数量$ k $的情况下。最后，我们表明，$ d $的素数元素集的关闭是$ \ hat {d} $的单位组的结合，而零件可忽略不计。
Let $D$ be the ring of $S$-integers in a global field and $\hat{D}$ its profinite completion. We discuss the relation between density in $D$ and the Haar measure of $\hat{D}$: in particular, we ask when the density of a subset $X$ of $D$ is equal to the Haar measure of its closure in $\hat{D}$. In order to have a precise statement, we give a general definition of density which encompasses the most commonly used ones. Using it we provide a necessary and sufficient condition for the equality between density and measure which subsumes a criterion due to Poonen and Stoll. In another direction, we extend the Davenport-Erdős theorem to every $D$ as above and offer a new interpretation of it as a "density=measure" result. Our point of view also provides a simple proof that in any $D$ the set of elements divisible by at most $k$ distinct primes has density 0 for any natural number $k$. Finally, we show that the closure of the set of prime elements of $D$ is the union of the group of units of $\hat{D}$ with a negligible part.
