Arithmetic structure of the exceptional set of projections

Changhao Chen; Zhengyan Miao

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arXiv · 2210.12677

Arithmetic structure of the exceptional set of projections

Changhao Chen·Zhengyan Miao

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Abstract

We study the arithmetic structure of the exceptional set of projections. For any bounded subset $E\subset \mathbb{R}^d$, let $$ \Omega=\{\xi\in \mathbb{R}: \dim_B(E+\xi E)=\dim_B E\}. $$ We prove that either $\Omega=\{0\}$ or $\Omega$ 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 \[ \overline{\dim}_B \, \{x+r_k\, y: (x, y)\in E\} <s. \]

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Changhao Chen, Zhengyan Miao. 2022-10-23. Arithmetic structure of the exceptional set of projections. https://arxiv.org/abs/2210.12677

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