  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.
