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. ]