Fourier Spectrum: Bridging Analysis and Geometry

• The Fourier spectrum is a continuous family of dimensions that quantifies the decay of Fourier transforms while interpolating between classical Fourier and Hausdorff dimensions.
• It leverages a one-parameter framework that defines energy measures and yields sharp criteria for convolution, sumset, and restriction problems in harmonic analysis.
• Its discrete representations and computational approaches enhance applications in signal recovery, additive combinatorics, and machine learning by providing precise analytic tools.

The Fourier spectrum refers to a family of dimensions associated to a measure or set in Euclidean space, capturing the asymptotic decay and distribution of its Fourier transform. Unlike the classical Fourier and Hausdorff dimensions—respectively measuring the pointwise polynomial decay and the geometric “size” via coverings—the Fourier spectrum introduces a one-parameter continuum of dimensions that interpolate between purely Fourier-analytic decay and geometric measure-theoretic size. This framework, explicitly formalized in recent works, provides new quantitative tools and sharp criteria for problems in harmonic analysis, fractal geometry, additive combinatorics, signal processing, and machine learning (Fraser, 2022, Carnovale et al., 2024, Carnovale et al., 2024).

1. Definition and Formulation of the Fourier Spectrum

Let $\mu$ be a finite Borel measure on $\mathbb{R}^d$. The Fourier transform is $$\widehat{\mu}(z) = \int e^{-2\pi i \langle z, x \rangle} d\mu(x)$$. For $s > 0$ and $\theta \in [0,1]$, the (s, θ)–energy is defined by

$$J_{s,\theta}(\mu) = \begin{cases} \sup_{z \in \mathbb{R}^d} |\widehat{\mu}(z)|^2 |z|^s, & \theta=0 \ \left( \int_{\mathbb{R}^d} |\widehat{\mu}(z)|^{2/\theta} |z|^{s/\theta-d} dz \right)^{\theta}, & \theta \in (0,1] \ \end{cases}$$

The θ–Fourier spectrum of μ is

$$\dim_F^{\theta} \mu = \sup \left\{ s \geq 0 : J_{s,\theta}(\mu) < \infty \right\}$$

For Borel sets $X \subset \mathbb{R}^d$, define

$$\dim_F^\theta X = \sup \left\{ \min\left\{ \dim_F^\theta \mu,\, d \right\} : \mathrm{supp}(\mu) \subset X,\, \mu\ \text{finite Borel} \right\}$$

Special cases:

• At $\theta=0$, this recovers the classical Fourier dimension (spectral decay rate):

$$\dim_F^0\mu = \sup\{s : |\widehat{\mu}(z)| \lesssim |z|^{-s/2} \} = \dim_F \mu$$

• At $\theta=1$, this is the Sobolev/energy dimension, which coincides with the Hausdorff dimension for Borel sets:

$$\dim_F^1 \mu = \sup \{ s : \int |\widehat{\mu}(z)|^2 |z|^{s-d} dz < \infty \} = \dim_S \mu$$

Thus, $\theta \mapsto \dim_F^\theta \mu$ interpolates between the Fourier and Sobolev/Hausdorff dimensions (Fraser, 2022, Carnovale et al., 2024, Carnovale et al., 2024).

2. Analytic Properties: Monotonicity, Concavity, and Bounds

The function $\theta \mapsto \dim_F^\theta \mu$ is non-decreasing, concave, and continuous on $(0,1]$. For all $\theta \in [0,1]$,

$\dim_F \mu \leq \dim_F^\theta \mu \leq \dim_S \mu$

For sets $X$, analogous interpolation holds between the Fourier and Hausdorff dimensions. The spectrum is stable under isometric embedding, concave via Hölder-type interpolation, and continuous (Lipschitz at $\theta = 0$ under Hölder control of $\widehat{\mu}$).

Sharp general bounds have been established (Carnovale et al., 2024): $\dim_F^\theta \mu \leq \dim_F \mu + \theta d$ with equality for Lebesgue measure and carefully constructed convolutions of Salem and zero–Fourier-dimension measures.

3. Discrete Representation via Fourier Coefficients

For measures $\mu$ supported on $[\varepsilon,1-\varepsilon]^d$, the spectrum can be expressed discretely in terms of Fourier coefficients

$$c_n = \widehat{\mu}(n) = \int e^{-2\pi i n \cdot x}\, d\mu(x),\quad n \in \mathbb{Z}^d$$

The precise equivalence is (Carnovale et al., 2024): $$J_{s,\theta}(\mu) \simeq_{d,\varepsilon,s,\theta} |c_0|^2 + \sum_{n \neq 0} |c_n|^2 |n|^{s - d\theta}$$ Thus

$$\dim_F^\theta \mu = \sup\{ s > 0 : \sum_{n \neq 0} |\widehat{\mu}(n)|^2 |n|^{s-d\theta} < \infty \}$$

This discrete perspective is powerful for computational analysis and for applications involving periodic and arithmetic structure.

4. Applications: Harmonic Analysis, Additive Combinatorics, and Geometry

The Fourier spectrum is effective in quantifying and solving several analytic and combinatorial problems:

• Convolution and Sumsets: The Fourier spectrum controls dimension growth under convolution. For finite Borel measures,

$$\dim_F^\theta(\mu * \nu) \geq \max_{\lambda \in [0,1]} \{ \dim_F^{\lambda\theta}\mu + \dim_F^{(1 - \lambda)\theta}\nu \}$$

Iteratively, for $k$-fold convolutions:

$$\dim_F^\theta(\mu^{*k}) = k \cdot \dim_F^{\theta/k} \mu$$

For sets $X,Y$, if $\dim_F^\lambda X > d - (1-\lambda)\dim_F Y$ for some $\lambda \in [0,1)$, then $X+Y$ has positive Lebesgue measure.

• Distance Set Problem: For Borel $X \subset \mathbb{R}^d$,
  • If $$\sup_{\theta \in [0,1]} \{ \dim_F^\theta X + \dim_F^{1-\theta} X \} > d$$, then the distance set $D(X)$ has positive 1-dimensional Lebesgue measure.
  • Otherwise,

$$\dim_H D(X) \geq 1-d + \sup_\theta \{ \dim_F^\theta X + \dim_F^{1-\theta} X \}$$

• Restriction Theory: The spectrum yields a continuum of Stein–Tomas type $L^q \to L^2$ Fourier restriction/extension estimates, superseding previous bounds based only on Fourier or Sobolev dimension. For example, for $\alpha = \dim_{\mathrm{Frost}}\mu$,

$$q > 2 + 2 \inf_{0 \leq \theta \leq 1,\, \dim_F^\theta \mu > \theta d} \frac{(d-\alpha)(2-\theta)}{\dim_F^\theta\mu - \alpha\theta}$$

guarantees the restriction estimate; the failure cases are also spectrum-sharp (Carnovale et al., 2024).

5. Explicit Calculations for Natural Measures

A wide variety of explicit formulas for $\dim_F^\theta \mu$ are available:

• Lebesgue on $[0,1]^d$: $\dim_F^\theta \mu = \theta d$ for all $\theta \in [0,1]$ (Carnovale et al., 2024).
• Riesz Products: For $\mu_{a,\lambda}$, a product measure, $\dim_F^\theta \mu_{a,\lambda}$ admits a closed formula involving the sequence parameters.
• Bernoulli Convolutions: For $p$-biased Cantor measure $\mu_p$,

$$\dim_F^\theta \mu_p = \theta - \theta \log_2 (1 + |2p - 1|^{2/\theta}),\quad \theta \in (0,1)$$

• Random Measures: If $$\mathbb{E}[|\widehat{\mu}(z)|^{2/\theta}] \lesssim |z|^{-s/\theta}$$, then almost surely $\dim_F^\theta \mu \geq s$ (Fraser, 2022).
• Fractional Brownian Images: The spectrum of the image of a set $Y$ under $B^\alpha : \mathbb{R}^n \to \mathbb{R}^k$ satisfies $$\dim_F^\theta B^\alpha(Y) \approx \min\{ k\theta, \dim_H Y / \alpha \}$$.

6. Spectrum in Signal Processing, Learning, and Computational Contexts

• Signal Recovery and Resolution: Exact Fourier spectrum recovery algorithms address the limitations of the DFT when true frequencies are off the sampling grid, by formulating and solving a nonlinear least squares problem to recover all frequency components and amplitudes (Andrecut, 2013).
• Fractional Periodicity Analysis: The fractional-period Fourier spectrum (FPS), and efficient FPS algorithms, are crucial for analyzing signals with “fractional” periodicities, as arise in structural biology and genomics. These operate by re-expressing the spectrum in terms of congruence-derivative sequences, substantially reducing computational costs compared to classical methods (Wang et al., 2016).
• Machine Learning (Boolean Functions): In computational learning theory, the “heavy” low-degree Fourier spectrum of Boolean functions underpins efficient PAC learning of DNF expressions and related function classes, enabling new polynomial or quasi-polynomial time algorithms by matching the spectrum within carefully controlled error (Feldman, 2012).
• Deep Generative Models: In image synthesis, architectures that explicitly enforce alignment between the Fourier spectrum of generated and real images achieve improved perceptual quality and spectral fidelity, traced by metrics such as the normalized power-spectrum-distance (Gargar, 2022).

7. Further Directions and Research Significance

The Fourier spectrum, as a refined analytic invariant, is central to new developments across real and harmonic analysis, discrete mathematics, and data-driven methodologies. It refines classical dimensional dichotomies, enables stability estimates under geometric and algebraic operations, and supplies exact or nearly exact criteria for phase transitions in convolution, sumset, and distance set problems (Fraser, 2022, Carnovale et al., 2024). Discrete and computational formulations (Carnovale et al., 2024, Wang et al., 2016) extend its reach into algorithmic settings of high practical relevance. Its intersection with spectral learning and signal processing (notably through new recoverability and expressivity criteria) maintains active relevance for modern data science, bridging abstract harmonic analysis with concrete applications.

Key Citations:

• "The Fourier spectrum and sumset type problems" (Fraser, 2022)
• "Obtaining the Fourier spectrum via Fourier coefficients" (Carnovale et al., 2024)
• "$L^2$ restriction estimates from the Fourier spectrum" (Carnovale et al., 2024)
• "Exact Fourier Spectrum Recovery" (Andrecut, 2013)
• "A Fast Algorithm for Computing the Fourier Spectrum of a Fractional Period" (Wang et al., 2016)
• "Learning DNF Expressions from Fourier Spectrum" (Feldman, 2012)
• "Explicit Use of Fourier Spectrum in Generative Adversarial Networks" (Gargar, 2022)