reference request – Is Fourier dimension finitely stable?


Let $A,B$ be compact subsets of $mathbb R$. Let $a=mathrm{dim}_F(A)$, $b=mathrm{dim}_F(B)$ be their Fourier dimensions, respectively. My questions are as follows:

  1. Is it true that $mathrm{dim}_F(Acup B)=max{a,b}$?

  2. If $A$ and $B$ are positively separated, is it true that $mathrm{dim}_F(Acup B)=max{a,b}$?

  3. In addition to 2, if $Asubseteq (0,1)$ and $Bsubseteq (2,3)$, is it true that $mathrm{dim}_F(Acup B)=max{a,b}$?

  4. In addition to 3, if $a=0$, is it true that $mathrm{dim}_F(Acup B)=b$?

I have searched online but I cannot find any references for that.