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/