Higher-order Fourier dimension and frequency decompositions

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Publication date: 13 August 2013

Abstract: This paper continues work begun in cite{M1}, in which we introduced a theory of Gowers uniformity norms for singular measures on

m a t h b b R^{d}

. There, given a

d

-dimensional measure

m u

, we introduced a

(k + 1) d

-dimensional measure

r i a n g l e^{k} m u

, and developed a Uniformity norm

| m u |_{U^{k}}

whose

2^{k}

-th power is equivalent to

r i a n g l e^{k} m u ([0, 1]^{d (k + 1)}

. In the present work, we introduce a fractal dimension associated to measures

m u

which we refer to as the

k

th-order Fourier dimension of

m u

. This

k

-th order Fourier dimension is a normalization of the asymptotic decay rate of the Fourier transform of the measure

i n t r i a n g l e^{k} m u (x; c d o t), d x

, 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

| m u - m u_{n} |_{U^{k}} i g h t a r r o w 0

, where

m u_{n}

is an approximation to the measure

m u

. This allows us to extract delicate information from the Fourier transform of a measure

m u

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.

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