论文标题
与二进制动力学系统中轨道相关的总和
Sums Associated with Orbits in the Binary Dynamical System
论文作者
论文摘要
1930年，G。H。Hardy和J. E. Littlewood得出的结果，涉及某些涉及共同体的差异率。在最近的术语中，可以根据单位圆圈旋转的动态系统中的轨道行为来解释其结果之一。现在，$ [0,1）$在基本$ 2 $中的数字扩展可以与其他动力学系统 - 二进制系统相关联。本文考虑了二进制系统中的轨道行为，与Hardy和Littlewood在涉及旋转的系统中实际上观察到的行为相对应。给定$ [0,1）$中的典型数字，其二进制数字的序列可以作为连续的，非空的有限块的无限顺序，每个块由所有零或所有块组成。这些块的长度之间的关系决定了与数字相关的强硬木材类型的行为类型。除其他结果外，在数字轨道中的点倒数的互换之和得出了上和下部估计值。这些估计是根据相关块的长度。对于上和下部估计值的必需“等效性”，发现了必要且充分的条件。 $ [0,1）$中的几乎所有数字都满足了此条件。
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.
