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- W213188538 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 $triangle^kmu$, and developed a Uniformity norm $|mu|_{U^k}$ whose $2^k$-th power is equivalent to $triangle^kmu([0,1]^{d(k+1)}$. 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 $int triangle^kmu(x;cdot),dx$, 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." @default.
- W213188538 created "2016-06-24" @default.
- W213188538 creator A5078494922 @default.
- W213188538 date "2013-08-13" @default.
- W213188538 modified "2023-09-27" @default.
- W213188538 title "Higher-order Fourier dimension and frequency decompositions" @default.
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