论文标题
线性形式系统的Duffin-Schaeffer猜想
The Duffin--Schaeffer conjecture for systems of linear forms
论文作者
论文摘要
我们将达芬（Duffin）的猜想扩展到$ n $变量中的$ m $线性表单系统的设置。也就是说，我们建立了一个标准，以确定在给定的近似速率上，几乎全部或几乎是$ n $ by- $ m $ m $线性形式系统的线性形式系统可以使用满足自然相关性条件的整数向量以这种速率近似。当$ m = n = 1 $时，这是经典的1941 Duffin--schaeffer猜想，这是Koukoulopoulos和Maynard在2020年证明的。 Pollington和Vaughan证明了更高维度的版本，其中$ M> 1 $和$ n = 1 $，1990年。我们在这里证明的一般性声明是由Beresnevich，Bernik，Dodson和Velani猜想的。对于没有任何共同点的近似值，他们还猜想了Catlin的猜想的广义版，并且在2010年，Beresnevich和Velani证明了$ M> 1美元的案例。 Catlin的经典猜想，其中$ m = n = 1 $，遵循经典的达芬 - schaeffer猜想。 $ M = 1 $和$ n> 1 $的剩余案例从我们的主要结果中随后。最后，通过质量转移原则，我们的主要结果暗示了他们的hausdorff测量类似物，这也由Beresnevich \ emph {et al}（2009）猜想。
We extend the Duffin--Schaeffer conjecture to the setting of systems of $m$ linear forms in $n$ variables. That is, we establish a criterion to determine whether, for a given rate of approximation, almost all or almost no $n$-by-$m$ systems of linear forms are approximable at that rate using integer vectors satisfying a natural coprimality condition. When $m=n=1$, this is the classical 1941 Duffin--Schaeffer conjecture, which was proved in 2020 by Koukoulopoulos and Maynard. Pollington and Vaughan proved the higher-dimensional version, where $m>1$ and $n=1$, in 1990. The general statement we prove here was conjectured in 2009 by Beresnevich, Bernik, Dodson, and Velani. For approximations with no coprimality requirement, they also conjectured a generalized version of Catlin's conjecture, and in 2010 Beresnevich and Velani proved the $m>1$ cases of that. Catlin's classical conjecture, where $m=n=1$, follows from the classical Duffin--Schaeffer conjecture. The remaining cases of the generalized version, where $m=1$ and $n>1$, follow from our main result. Finally, through the Mass Transference Principle, our main results imply their Hausdorff measure analogues, which were also conjectured by Beresnevich \emph{et al} (2009).
