Higher-order Fourier dimension and frequency decompositions

Abstract

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 ${△}^k\mu$, and developed a Uniformity norm $\Vert \mu {\Vert }_{U^k}$ whose $2^k$-th power is equivalent to ${△}^k\mu ([0,1{]}^{d(k+1)}$. 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 $\int {△}^k\mu (x;\cdot )dx$, 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 $\Vert \mu -{\mu }_n{\Vert }_{U^k}\to 0$, 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.

References in corpus (2)

• Long progressions in sets of fractional dimension
• Gowers norms for singular measures

Cited by in corpus (2)

• Long progressions in sets of fractional dimension
• A differentiation theorem for uniform measures