On the Fourier dimension of fractional Brownian graphs

TL;DR

This work proves that the Fourier dimension of the graph $G(B)$ of a fractional Brownian motion with Hurst parameter $H\in(0,1/2)$ is almost surely $1$, resolving a conjecture of Fraser and Sahlsten. The authors reduce the problem to sharp horizontal moment bounds for the graph measure's Fourier transform by exploiting the Kahane-type moment decomposition $\mathbb E[|\hat{\mu}_G|^{2q}]$ and recasting the computation as integrals $\mathcal{I}[\varepsilon, G_\varepsilon]$ over the simplex. A central contribution is an intricate integration-by-parts and combinatorial framework that controls these multi-dimensional integrals via the covariance structure of the process, yielding a bound $\mathbb E[|\hat{\mu}_G(\xi_1,\xi_2)|^{2q}] \lesssim |\xi_1|^{-q}$. Combined with known vertical bounds for the image measure, this establishes $\dim_{\rm F} G(B)=1$ and, since $\dim_{\rm H} G(B)=2-H>1$, provides the exact gap between Hausdorff and Fourier dimensions for the graph. The result completes the Salem-set classification for these graphs and has implications for the arithmetic and geometric properties 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 $\dim_{\rm H}(G(B)) - \dim_{\rm F}(G(B))$ 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.

Figures (2)

• Figure 1: Example of values for the variables $\varepsilon$ and $\sigma$, towards an integration in $\Delta_{\alpha,T}$
• Figure 2: Example of action in the 6-dimensional partial simplex $\Delta_{\alpha,T}$ where $\alpha = \sigma_{1}\sigma_{5}(\varepsilon)$.

Theorems & Definitions (22)

• Theorem 1.1
• Definition 2.1
• Proposition 2.2
• Lemma 2.3
• Definition 2.4
• Example 2.5
• Lemma 2.6
• Lemma 2.7
• Example 2.8
• Lemma 2.9