论文标题
Khintchine的定理的理由来自不同地方的社区
Khintchine's Theorem with rationals coming from neighborhoods in different places
论文作者
论文摘要
达芬（Duffin） - 施弗（Schaeffer）的猜想回答了一个问题，即一个人如何以降低形式（强加条件）的理性数字近似非理性的问题，其中近似的准确性取决于有理数。它可以看作是Khintchine定理的类似物，其额外限制仅允许以简化形式进行理性。也强加了其他条件，例如分子或分母的素数，无方整数或特定算术进程等的元素，并研究了Khintchine定理的类似物。我们证明了Khintchine定理的版本，其中有理数是从$ \ Mathbb {q} $完成的球中采购的（即欧几里得或$ p $ -ADIC），而近似值则以独特的第二完成。最后，通过将质量转移原理用于Hausdorff措施，我们能够将结果扩展到其相应的类似物中，而HAAR测量取代了Hausdorff措施，从而建立了Jarník定理的类似物。
The Duffin--Schaeffer Conjecture answers a question on how well one can approximate irrationals by rational numbers in reduced form (an imposed condition) where the accuracy of the approximation depends on the rational number. It can be viewed as an analogue to Khintchine's Theorem with the added restriction of only allowing rationals in reduced form. Other conditions such as numerator or denominator a prime, a square-free integer, or an element of a particular arithmetic progression, etc. have also been imposed and analogues of Khintchine's Theorem studied. We prove versions of Khintchine's Theorem where the rational numbers are sourced from a ball in some completion of $\mathbb{Q}$ (i.e. Euclidean or $p$-adic), while the approximations are carried out in a distinct second completion. Finally, by using a mass transference principle for Hausdorff measures, we are able to extend our results to their corresponding analogues with Haar measures replaced by Hausdorff measures, thereby establishing an analogue of Jarník's Theorem.
