For a specially constructed log-correlated field on T^d, the GMC measure almost surely has Fourier dimension d-γ^2 when γ<√(2d)/2 and (√(2d)-γ)^2 when √(2d)/2<=γ<√(2d), for all d>=1.
Multifractal analysis of Gaussian multiplicative chaos and applications
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Exact values of Fourier dimensions of Gaussian multiplicative chaos on high dimensional torus
For a specially constructed log-correlated field on T^d, the GMC measure almost surely has Fourier dimension d-γ^2 when γ<√(2d)/2 and (√(2d)-γ)^2 when √(2d)/2<=γ<√(2d), for all d>=1.