Arithmetic structure of the exceptional set of projections

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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