Borel sets with Fourier dimension at least 2 have distance sets of full Hausdorff dimension in any ambient dimension d, and sets with Fourier spectrum at least d/4 + 1 at theta = 1/2 also achieve this even when their Fourier dimension is zero provided d is at least 4.
Mattila, Fourier analysis and Hausdorff dimension, Cambridge Studies in Advanced Mathematics
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On Fourier decay and the distance set problem
Borel sets with Fourier dimension at least 2 have distance sets of full Hausdorff dimension in any ambient dimension d, and sets with Fourier spectrum at least d/4 + 1 at theta = 1/2 also achieve this even when their Fourier dimension is zero provided d is at least 4.