论文标题
与平面和一般组的交集的Hausdorff尺寸
Hausdorff dimension of intersections with planes and general sets
论文作者
论文摘要
我们给出一般家庭的条件$p_λ：\ r^n \至\ r^m，λ\inλ，正交预测的$，保证Hausdorff dimension公式$ \ dim a \ capp_λ^{ - 1} \ capp_λ^{ - 1} \ {u \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ s-m $ subs $ subs $ subsets $ subsets $ a \ a \ a \ a a \ rn $ s $二维的Hausdorff度量，$ s> m $，并且密度为正。 As an application we prove for measurable sets $A,B\subset\Rn$ with positive $s$- and $t$-dimensional measures, and with positive lower density that if $s + (n-1)t/n > n$, then $\dim A\cap (g(B)+z) = s+t - n$ for almost all rotations $g$ and for positively many $z\in\Rn$.
We give conditions on a general family $P_λ:\R^n\to\R^m, λ\in Λ,$ of orthogonal projections which guarantee that the Hausdorff dimension formula $\dim A\cap P_λ^{-1}\{u\}=s-m$ holds generically for measurable sets $A\subset\Rn$ with positive and finite $s$-dimensional Hausdorff measure, $s>m$, and with positive lower density. As an application we prove for measurable sets $A,B\subset\Rn$ with positive $s$- and $t$-dimensional measures, and with positive lower density that if $s + (n-1)t/n > n$, then $\dim A\cap (g(B)+z) = s+t - n$ for almost all rotations $g$ and for positively many $z\in\Rn$.
