论文标题
Dirichlet不仅很糟糕，而且很奇怪
Dirichlet is not just Bad and Singular
论文作者
论文摘要
众所周知，在尺寸的一组中，Dirichlet可改进的实数完全由近似和奇异的数字组成。我们表明，在较高的维度上，事实并非如此，证明存在许多dirichlet的即将改进的向量，它们既不是近似也不是奇异的。这是一个更强有力的陈述的结果，涉及非常相似的点。在最后一节中，我们制定了中间迪里奇莱特（Dirichlet）的概念，即通过每个中间维度的理性平面进行近似，并表明它们是重合的。这自然扩展了Davenport和Schmidt（1969）的经典定理，该定理指出，Dirichlet定理的同时形式是可以改进的。因此，我们的主要“连续性”结果对于相应的中间毒液剂组同样有效，这些中间位于不良，可近似和dircihlet的改进点。
It is well known that in dimension one the set of Dirichlet improvable real numbers consists precisely of badly approximable and singular numbers. We show that in higher dimensions this is not the case by proving that there exist continuum many Dirichlet improvable vectors that are neither badly approximable nor singular. This is a consequence of a stronger statement that involves very well approximable points. In the last section we formulate the notion of intermediate Dirichlet improvable sets concerning approximations by rational planes of every intermediate dimension and show that they coincide. This naturally extends a classical theorem of Davenport and Schmidt (1969) which states that the simultaneous form of Dirichlet's theorem is improvable if and only if the dual form is improvable. Consequently, our main "continuum" result is equally valid for the corresponding intermediate Diophantine sets of badly approximable, singular and Dircihlet improvable points.
