摘要
PaulErd s曾提出如下关于实直线R的问题 :是否对R的每一个无限子集X ,都存在一个具有正测度 (Lebesgue测度 )的闭子集E ,使得E的任何子集都不相似于X(E的任何子集都不与X线性同胚 ) .1 984年 ,Falconer证明了如下结论 :对于一个满足limxn =0和limxn+1xn =1的单调递减的正实数列 {xn},Erd s问题有一个部分肯定的解答 .本文将证明 :上述关于数列的条件可以替换为更一般的 (弱一些的 )条件 .最后把本文的相应结论推广到有限维欧氏空间Rn 中 .
Paul Erds once posed the following problem about real line R: is it true that, for every infinite set X, there is a closed set E with positive lebesgue measure such that E doesn't contain any subset similar to X (i.e., there is no subset of E, which is a linear homeomorphic image of X). In 1984, K. J. Falconer proved the following: for a decreasing sequence of positive numbers {x n} such that \%lim\% x n=0 and \%lim\%x n+1 x n=1, Erds problem has a partial positive answer. This paper will prove that: the requirement for the sequence can be replaced by a more general (weaker) requirement. Finally we will generalize corresponding result to n dimension Euclidean space.
