The set L of all Liouville numbers is an example of a 0-dimensional Salem set such that its sum-set equals the whole real line, that is, L+L=R. Thus,
dimF(L) = 0 and dimF( L+L) =1.
For the construction of a Rajchman measure over L see the paper of Bluhm. For the proof of L+L=R see the paper of Erdős.