On the Fourier coefficients of critical Gaussian multiplicative chaos

We continue the study of the Fourier coefficients of Gaussian multiplicative chaos (GMC) recently initiated by Garban and Vargas. We show that if $\{c_n{\}}_{n\ge 1}$ are the Fourier coefficients of critical GMC on the unit interval, then $(\log n{)}^{\alpha }{c}_n$ converges to zero in probability as $n$ tends to infinity for any $\alpha <1\mathrm{/}4$.