关于丢番图集类的和集与笛卡尔积的度量结果
Metric results on sumsets and Cartesian products of classes of Diophantine sets
AI总结：
受Erdős关于刘维尔数和积结果启发，研究指定无理性指数实数类和集的度量性质，证明其和集勒贝格测度下几乎覆盖全体实数，补集豪斯多夫维数小，还拓展到笛卡尔积维数、缺数字康托尔集填充维数等问题，推广了Erdős的方法。
摘要：
Erdős（埃尔德什）证明了任意实数都可以表示为两个Liouville（刘维尔）数的和与积。受这些结果启发，我们研究具有指定（或有界）无理性指数的实数类的和集。我们证明这类和集通常规模很大：实际上，关于Lebesgue（勒贝格）测度的几乎所有实数，都可以表示为两个具有足够大指定无理性指数的数之和。事实上，其补集的Hausdorff（豪斯多夫）维数很小；即便我们对有理逼近的阶数施加更精细的条件（关于逼近函数的“精确逼近”），该结果仍然成立。作为应用，我们证明在许多情况下，具有指定无理性指数的集合的笛卡尔积的Hausdorff维数超过预期维数，即单个集合Hausdorff维数之和。我们还研究了它们的packing（填充）维数。当限制在经典的缺数字Cantor（康托尔）集上、相对于其自然Cantor测度时，类似结论同样成立。特别地，我们证明了具有指定大无理性指数的数的子集具有满填充维数，即与整个Cantor集的填充维数相同。这补充了关于这些集合Hausdorff维数的部分结果，而该问题是丢番图逼近中被广泛研究的课题。我们的证明基于Erdős的思想，但对其进行了大幅拓展。
英文摘要：
Erdős proved that any real number can be written as a sum, and a product, of two Liouville numbers. Motivated by these results, we study sumsets of classes of real numbers with prescribed (or bounded) irrationality exponents. We show that such sumsets turn out to be large in general, indeed almost every real number with respect to Lebesgue measure can be written as the sum of two numbers with sufficiently large prescribed irrationality exponents. In fact the Hausdorff dimension of the complement is small, and the result remains true if we impose considerably refined conditions on the orders of rational approximation (``exact approximation'' with respect to an approximation function). As an application, we show that in many cases the Hausdorff dimension of Cartesian products of sets with prescribed irrationality exponent exceeds the expected dimension, that is the sum of the single Hausdorff dimensions. We also address their packing dimensions. Similar results hold when restricting to classical missing digit Cantor sets, relative to its natural Cantor measure. In particular, we prove that the subset of numbers with prescribed large irrationality exponent has full packing dimension, i.e. the same packing dimension as the entire Cantor set. This complements some results on the Hausdorff dimension of these sets, which is an extensively studied topic in Diophantine approximation. Our proofs are based on ideas of Erdős, but vastly extend them.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
