一类康托型集合的子集的维数

Dimensions of Subsets of a Class of Cantor-type Sets

设Ｅ为实直线上一康托型集， Ｅα＝ Ｅ ＋ α＝｛β＋α：β∈Ｅ｝；－１≤α≤ １．设 Ｇｐ＝｛β∈Ｅ－Ｅ： ｄｉｍＨ（Ｅα Ｅ）＝ｄｉｍＢ（Ｅα Ｅ）＝ｐｄｉｍＨ Ｅ）， ０＜ ｐ＜ １，此处Ｅ－Ｅ＝｛ｘ－ｙ：ｘ，ｙ Ｅ｝．在一定的条件下，集合Ｅα Ｅ与Ｇｐ的分形维数被确定．

Let E be a Cantor-type set on real line and let Ea = E+a = { } for . Let Go = { ; dimH(E E) = dimB(Ea E) = pdimH E}, 0 < p < 1, where E - E = {x - y: x,y E}. The fractal dimensions of Ea E and Gp are determined under some conditions.