Higher-order Fourier dimension and frequency decompositions

This paper continues work begun in \cite{M1}, in which we introduced a theory of Gowers uniformity norms for singular measures on $\mathbb{R}^d$. There, given a $d$-dimensional measure $\mu$, we introduced a $(k+1)d$-dimensional measure $\triangle^k\mu$, and developed a Uniformity norm $\|\mu\|_{U^k}$ whose $2^k$-th power is equivalent to $\triangle^k\mu([0,1]^{d(k+1)}$. In the present work, we introduce a fractal dimension associated to measures $\mu$ which we refer to as the $k$th-order Fourier dimension of $\mu$. This $k$-th order Fourier dimension is a normalization of the asymptotic decay rate of the Fourier transform of the measure $\int \triangle^k\mu(x;\cdot)\,dx$, and coincides with the classic Fourier dimension in the case that $k=1$. It provides quantitative control on the size of the $U^k$ norm. The main result of the present paper is that this higher-order Fourier dimension controls the rate at which $\|\mu-\mu_n\|_{U^k}\rightarrow 0$, where $\mu_n$ is an approximation to the measure $\mu$. This allows us to extract delicate information from the Fourier transform of a measure $\mu$ and the interactions of its frequency components, which is not available from the $L^p$ norms- or the decay- of the Fourier transform. In future work \cite{M4}, we apply this to obtain a differentiation theorem for singular measures.