# 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.