在贝西科维奇的合奏中

摘要：

下面是一个由方向覆盖一个角的线段组成的消失区平面集的例子：设E是剖分比为1/4的康托尔集（因此尺寸1/2）由水平轴承载，E'是E通过比率2的位似变换的图像，其中心不在水平轴上;连接E和E'的线段的并集是所讨论的集合。这个例子是作者在1969年用错误的证明给出的。本文包含一个简短的证明基础上的投影定理贝西科维奇，和长期的证明导致调查的算术，几何和分析性质的水平部分的一套。这项研究复制了理查德·凯尼恩对类似问题的研究。在平面和多维空间中提供了结构的变型。

英文摘要：

Here is an example of a plane set of vanishing area and consisting of line-segments whose directions cover an angle : let E be a Cantor set of dissection ratio 1/4 (therefore dimension 1/2) carried by the horizontal axis and E' the image of E by an homothetic transformation of ratio 2 whose center is not on the horizontal axis ; the union of the line segments joining E and E' is the set in question. The example was given in 1969 by the author with a wrong proof. The present article contains a short proof based on the projection theorem of Besicovitch, and a long proof resulting from the investigation of the arithmetic, geometric and analytic properties of the horizontal sections of the set. This investigation copies the study by Richard Kenyon of a similar problem. Variations of the construction are provided in the plane and in multidimensional spaces.

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。