论文标题
特殊预测集的算术结构
Arithmetic structure of the exceptional set of projections
论文作者
论文摘要
我们研究了一系列特殊的投影集的算术结构。对于任何有限的子集$ e \ subset \ mathbb {r}^d $，让$$ω= \ {ξ\ in \ mathbb {r}：\ dim_b（e+ξe）= \ dim_b e \ \}。 $$我们证明$ω= \ {0 \} $或$ω$是$ \ mathbb {r} $的子场。我们表明，通常，该语句不适合Hausdorff维度和较低的框尺寸。此外，对于任何$ s \ in（0，1] $和一个序列$（r_k）\ subset \ mathbb {r} $，我们构造了ahlfors $ s $ s $ regular Set $ e \ subset \ subset \ subset \ subset \ mathbb {r}^2 $ \ edline {\ dim} _b \，\ {x+r_k \，y ：（ x，y）\在e \} <s中。
We study the arithmetic structure of the exceptional set of projections. For any bounded subset $E\subset \mathbb{R}^d$, let $$ Ω=\{ξ\in \mathbb{R}: \dim_B(E+ξE)=\dim_B E\}. $$ We prove that either $Ω=\{0\}$ or $Ω$ is a subfield of $\mathbb{R}$. We show that in general the statement does not hold for Hausdorff dimension and lower box dimension. Moreover, for any $s\in (0, 1]$ and a sequence $(r_k) \subset \mathbb{R}$, we construct a Ahlfors $s$-regular set $E\subset \mathbb{R}^2$ such that for any $r_k, k\in \mathbb{N}$, we have \[ \overline{\dim}_B \, \{x+r_k\, y: (x, y)\in E\} <s. \]
