与二元动力系统中轨道相关的和
Sums Associated with Orbits in the Binary Dynamical System
AI总结：
本文研究了二元动力系统中对应Hardy-Littlewood观察到的旋转系统行为的轨道特性，基于二进制数分块长度导出了轨道点倒数幂和的上下界估计，并找到了该估计本质等价的充要条件。
摘要：
1930年，G. H. Hardy和J. E. Littlewood推导出了关于涉及余割的某些级数发散速度的结果。用更现代的术语来说，他们的一个结果可以从单位圆上旋转的动力系统中轨道行为的角度来解释。现在，将[0,1)中的数以2为底展开可以与另一个不同的动力系统——二元系统相关联。本文考虑二元系统中对应于Hardy和Littlewood在涉及旋转的系统中实际观察到的行为的轨道行为。给定[0,1)中的一个典型数，其二进制数字序列可以排列为由连续的、非空的有限块组成的无限序列，每个块全由0或全由1组成。这些块长度之间的关系决定了与该数相关联的Hardy-Littlewood类型行为。除其他结果外，本文还导出了该数轨道中各点倒数幂的和的上下界估计。这些估计以相关联的块的长度表示。本文还找到了上下界估计本质“等价”的充分必要条件。[0,1)中的几乎所有数都满足该条件。
英文摘要：
In 1930, G. H. Hardy and J. E. Littlewood derived results concerning rates of divergence of certain series involving cosecants. In more recent terminology, one of their results can be interpreted in terms of the behaviour of orbits in a dynamical system that is a rotation on the unit circle. Now, the expansion of numbers in $[0,1)$ to the base $2$ can be associated with a different dynamical system -- the binary system. This article considers orbit behaviour in the binary system that corresponds to the behaviour that was, in effect, observed by Hardy and Littlewood in systems involving rotations. Given a typical number in $[0,1)$, the sequence of its binary digits may be arranged as an infinite sequence of consecutive, non-empty, finite blocks, each block consisting of all zeros or all ones. The relationships between the lengths of these blocks determine Hardy-Littlewood types of behaviour associated with the number. Amongst other results, upper and lower estimates are derived for the sums of powers of the reciprocals of points in the orbit of the number. These estimates are in terms of the lengths of the associated blocks. A necessary and sufficient condition is found for the essential `equivalence' of the upper and lower estimates. Almost all numbers in $[0,1)$ satisfy this condition.
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