Arithmetic structure of the exceptional set of projections

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