On the Fourier dimension of fractional Brownian graphs

Abstract

In this note we prove that the Fourier dimension of the graph $G(B)$ of a fractional Brownian motion $B$ with Hurst parameter $H\in (0,1/2)$ is equal to 1. This finishes to solve a conjecture by Fraser and Sahlsten. It also yields an exact formula for the gap ${\dim }_{\mathrm{H}}(G(B))-{\dim }_{\mathrm{F}}(G(B))$ between the Hausdorff dimension and the Fourier dimension of $G(B)$. The proof is based on an intricate combinatorics procedure for multiple integrals related to the covariance function of the fractional Brownian motion.