论文标题
中间$α$ cantor套件与固定翻译的交叉点
Intersections of middle-$α$ Cantor sets with a fixed translation
论文作者
论文摘要
对于$λ\ in（0,1/3] $让$c_λ$为中间 - $（1-2λ）$ cantor设置为$ \ mathbb r $。给定$ t \ in [-1,1] $ in [-1,1] $，不包括琐事，我们显示了\ [λ（t）：= = = = = \ weft \ weft \ weft \ {λ\ in（0,1/3]：0,0,1/3]： C_λ\ CAP（C_λ+T）\ ne \ neveryset \ right \} \]是一个拓扑室，尤其是lebesgue量和完整的hausdorff尺寸，我们尤其是$λ（t）$。 λ_β（t）：= \ left \ {λ\inλ（t）：\ dim_h（c_λ\ cap（c_λ+t））= \ dim_p（c_λ\ cap（c_λ+t）=β\β\ frac \ frac {\ frac {\ log 2}} $（-β\logβ-（1-β）\ log \ frac {1-β} {2}）/\ log 3 $。
For $λ\in(0,1/3]$ let $C_λ$ be the middle-$(1-2λ)$ Cantor set in $\mathbb R$. Given $t\in[-1,1]$, excluding the trivial case we show that \[ Λ(t):=\left\{λ\in(0,1/3]: C_λ\cap(C_λ+t)\ne\emptyset\right\} \] is a topological Cantor set with zero Lebesgue measure and full Hausdorff dimension. In particular, we calculate the local dimension of $Λ(t)$, which reveals a dimensional variation principle. Furthermore, for any $β\in[0,1]$ we show that the level set \[ Λ_β(t):=\left\{λ\inΛ(t): \dim_H(C_λ\cap(C_λ+t))=\dim_P(C_λ\cap(C_λ+t))=β\frac{\log 2}{-\log λ}\right\} \] has equal Hausdorff and packing dimension $(-β\logβ-(1-β)\log\frac{1-β}{2})/\log 3$. We also show that the set of $λ\inΛ(t)$ for which $\dim_H(C_λ\cap(C_λ+t))\ne\dim_P(C_λ\cap(C_λ+t))$ has full Hausdorff dimension.
