平面直线的维数谱猜想
The Dimension Spectrum Conjecture for Planar Lines
摘要：
设$L_{a,b}$是欧几里得平面上的一条直线，斜率为$a$，截距为$b$。维数谱$\spec(L_{a,b})$是$L_{a,b}$上各个点的所有有效维数的集合。维数谱猜想指出，对于每一条线$L_{a,b}$，$L_{a,b}$的谱包含一个单位区间。本文证明了维数谱猜想成立。令$(a,b)$为斜率-截距对，并令$d = \min\{\dim(a,b), 1\}$。对于每个$s \in (0, 1)$，我们构造一个点$x$，使得$\dim(x, ax + b) = d + s$。因此，我们证明了$\spec(L_{a,b})$包含区间$(d, 1+ d)$。Turetsky、Lutz和Stull的结果表明，$\spec(L_{a,b})$含有端点$d$和$1+d$。对于每个平面线$L_{a,b}$，合计为$[d, 1 + d] \subseteq \spec(L_{a,b})$。
英文摘要：
Let $L_{a,b}$ be a line in the Euclidean plane with slope $a$ and intercept $b$. The dimension spectrum $\spec(L_{a,b})$ is the set of all effective dimensions of individual points on $L_{a,b}$. The dimension spectrum conjecture states that, for every line $L_{a,b}$, the spectrum of $L_{a,b}$ contains a unit interval. In this paper we prove that the dimension spectrum conjecture is true. Let $(a,b)$ be a slope-intercept pair, and let $d = \min\{\dim(a,b), 1\}$. For every $s \in (0, 1)$, we construct a point $x$ such that $\dim(x, ax + b) = d + s$. Thus, we show that $\spec(L_{a,b})$ contains the interval $(d, 1+ d)$. Results of Turetsky , and Lutz and Stull, show that $\spec(L_{a,b})$ contain the endpoints $d$ and $1+d$. Taken together, $[d, 1 + d] \subseteq \spec(L_{a,b})$, for every planar line $L_{a,b}$.
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