arXiv subjects

Marc Carnovale

Publications and source records attributed to Marc Carnovale.

8 recordsLinked to original sources

arXiv2026

Fourier restriction estimates based on $L^q$-dimensions: beyond Stein--Tomas

The well-known Stein--Tomas restriction theorem gives the sharp range of $p$ for which $L^p\to L^2$ restriction estimates hold for the surface measure on the sphere. This was generalised to arbitrary measures satisfying certain Fourier decay and Frostman conditions by Mockenhaupt, Mitsis, and Bak--Seeger, with the most general version now a fundamental result in harmonic analysis. The Frostman condition essentially asks for uniform control on the measure of small balls and is the endpoint of a continuum of more nuanced conditions which describe the local fluctuations of the measure. This analysis gives rise to the $L^q$-dimensions of a measure and these are a central concept in fractal geometry and a crucial tool in multifractal analysis and the theory of large deviations. In this paper we prove a new Fourier restriction theorem which uses the $L^q$-dimensions instead of the Frostman condition, thus providing a continuum of estimates which recover Stein--Tomas at the endpoint. Our proof gives the endpoint estimate for all values of $q\in(1,\infty]$ via Stein's complex interpolation. In particular, in the case $q=\infty$ this partially resolves a question raised by Bak and Seeger. We explore when our theorem improves on Stein--Tomas, that is, when the range is not optimised at $q=\infty$, and show that this is the case quite generally, including for certain Mandelbrot cascade measures and measures with multifractal behaviour. On the way to proving our main theorem we obtain a novel description of the $L^q$-dimensions based on certain convolution norms, which may be of interest in its own right.

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arXiv2026

On the number of 3APs in fractal sets

We use techniques from the study of the Falconer distance conjecture to explore conditions which guarantee largeness (in terms of bounded $L^2$ density/Lebesgue measure and Hausdorff measure) of the set of lengths of step-sizes of three-term arithmetic progressions which occur within fractal sets, as well as analogous statements in discrete settings. Our main result is a version of {\L}aba and Pramanik's result in arxiv:0712.3882 that relies only on an assumption of a lower bound, $\delta$, on the mass of the measure $\mu$ together with an upper bound, $M$ on the $L^q$ norm of its Fourier transform for some $q\in(2,3]$ depending on the parameters $\delta$ and $M$.

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arXiv2024

$L^2$ restriction estimates from the Fourier spectrum

The Stein--Tomas restriction theorem is an important result in Fourier restriction theory. It gives a range of $q$ for which $L^q\to L^2$ restriction estimates hold for a given measure, in terms of the Fourier and Frostman dimensions of the measure. We generalise this result by using the Fourier spectrum; a family of dimensions that interpolate between the Fourier and Sobolev dimensions for measures. This gives us a continuum of Stein--Tomas type estimates, and optimising over this continuum gives a new $L^q\to L^2$ restriction theorem which often outperforms the Stein--Tomas result. We also provide results in the other direction by giving a range of $q$ in terms of the Fourier spectrum for which $L^q\to L^2$ restriction estimates fail, generalising an observation of Hambrook and {\L}aba. We illustrate our results with several examples, including the surface measure on the cone, the moment curve, and several fractal measures.

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arXiv2024

Obtaining the Fourier spectrum via Fourier coefficients

The Fourier spectrum is a family of dimensions that interpolates between the Fourier and Hausdorff dimensions and are defined in terms of certain energies which capture Fourier decay. In this paper we obtain a convenient discrete representation of those energies using the Fourier coefficients. As an example application, we use this representation to establish sharp bounds for the Fourier spectrum of a general measure with bounded support, improving previous estimates of the second-named author

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arXiv2015

Gowers norms for singular measures

Gowers introduced the notion of uniformity norm $\|f\|_{U^k(G)}$ of a bounded function $f:G\rightarrow\mathbb{R}$ on an abelian group $G$ in order to provide a Fourier-theoretic proof of Szemeredi's Theorem, that is, that a subset of the integers of positive upper density contains arbitrarily long arithmetic progressions. Since then, Gowers norms have found a number of other uses, both within and outside of Additive Combinatorics. The $U^k$ norm is defined in terms of an operator $\triangle^k : L^{\infty}(G)\mapsto L^{\infty} (G^{k+1})$. In this paper, we introduce an analogue of the object $\triangle^k f$ when $f$ is a singular measure on the torus $\mathbb{T}^d$, and similarly an object $\|μ\|_{U^k}$. We provide criteria for $\triangle^k μ$ to exist, which turns out to be equivalent to finiteness of $\||μ|\|_{U^k}$, and show that when $μ$ is absolutely continuous with density $f$, then the objects which we have introduced are reduced to the standard $\triangle^kf$ and $\|f\|_{U^k(\mathbb{T})}$. We further introduce a higher-order inner product between measures of finite $U^k$ norm and prove a Gowers-Cauchy-Schwarz inequality for this inner product.

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arXiv2015

Higher-order Fourier dimension and frequency decompositions

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 $μ$, we introduced a $(k+1)d$-dimensional measure $\triangle^kμ$, and developed a Uniformity norm $\|μ\|_{U^k}$ whose $2^k$-th power is equivalent to $\triangle^kμ([0,1]^{d(k+1)}$. In the present work, we introduce a fractal dimension associated to measures $μ$ which we refer to as the $k$th-order Fourier dimension of $μ$. This $k$-th order Fourier dimension is a normalization of the asymptotic decay rate of the Fourier transform of the measure $\int \triangle^kμ(x;\cdot)\,dx$, 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 $\|μ-μ_n\|_{U^k}\rightarrow 0$, where $μ_n$ is an approximation to the measure $μ$. This allows us to extract delicate information from the Fourier transform of a measure $μ$ 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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arXiv2015

Long progressions in sets of fractional dimension

We demonstrate $k+1$-term arithmetic progressions in certain subsets of the real line whose "higher-order Fourier dimension" is sufficiently close to 1. This Fourier dimension, introduced in previous work, is a higher-order (in the sense of Additive Combinatorics and uniformity norms) extension of the Fourier dimension of Geometric Measure Theory, and can be understood as asking that the uniformity norm of a measure, restricted to a given scale, decay as the scale increases. We further obtain quantitative information about the size and $L^p$ regularity of the set of common distances of the artihmetic progressions contained in the subsets of $\mathbb{R}$ under consideration.

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arXiv2013

A differentiation theorem for uniform measures

Using the notion of higher-order Fourier dimension introduced in \cite{M2} (which was a sort of psuedorandomness condition stemming from the Gowers norms of Additive Combinatorics), we prove a maximal theorem and corresponding differentiation theorem for singular measures on $\R^d$, $d=1,2,...$. This extends results begun by Hardy and Littlewood for balls in $\R^d$ and continued by Stein \cite{stein} for spheres in $\R^{d\geq 3}$ and Bourgain for circles in $\R^2$, first considered for more general spaces in \cite{rubio}, and shown to hold for some singular subsets of the reals for the first time in \cite{LabaDiff}. Notably, unlike the more delicate of the previous results on differentiation such as \cite{Bourgain} and \cite{LabaDiff}, the assumption of higher-order Fourier dimension subsumes all of the geometric or combinatorial input necessary for one to obtain our theorem, and suggests a new approach to some problems in Harmonic Analysis.

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