There is no constraint on the relation between the Fourier and Hausdorff dimension of a set beyond the condition that the Fourier dimension must not exceed the Hausdorff dimension.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-sm206-1-3, author = {Thomas William K\"orner}, title = {Hausdorff and Fourier dimension}, journal = {Studia Mathematica}, volume = {204}, year = {2011}, pages = {37-50}, zbl = {1240.42017}, language = {en}, url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm206-1-3} }
Thomas William Körner. Hausdorff and Fourier dimension. Studia Mathematica, Tome 204 (2011) pp. 37-50. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm206-1-3/