There is no non-trivial constraint on the Hausdorff dimension of sums of a set with itself.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-sm188-3-4, author = {T. W. K\"orner}, title = {Hausdorff dimension of sums of sets with themselves}, journal = {Studia Mathematica}, volume = {187}, year = {2008}, pages = {287-295}, zbl = {1200.28007}, language = {en}, url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm188-3-4} }
T. W. Körner. Hausdorff dimension of sums of sets with themselves. Studia Mathematica, Tome 187 (2008) pp. 287-295. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm188-3-4/