On the Fourier dimension and a modification

Ekström, Fredrik; Persson, Tomas; Schmeling, Jörg

doi:10.4171/jfg/23

Cited by 23 publications

(25 citation statements)

References 13 publications

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“…Theorem 2.5 leads to a bound on the rate of decay of the Fourier transform µ of µ, or, equivalently, on the Fourier dimension of the measure defined as the supremum value of s such that | µ(ξ)| ≤ C|ξ| −s/2 (ξ ∈ R 2 ) for some constant C; see [8,27] for recent discussions on Fourier dimensions. One might conjecture that, as is fairly typical for random measures, the Fourier dimension of the LQG measure equals its Hausdorff dimension for all 0 < γ < 2 − √ 2.…”

Section: 2mentioning

confidence: 99%

Exact dimensionality and projection properties of Gaussian multiplicative chaos measures

Falconer

Jin

2019

Trans. Amer. Math. Soc.

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Given a measure ν on a regular planar domain D, the Gaussian multiplicative chaos measure of ν studied in this paper is the random measure ν obtained as the limit of the exponential of the γ-parameter circle averages of the Gaussian free field on D weighted by ν. We investigate the dimensional and geometric properties of these random measures. We first show that if ν is a finite Borel measure on D with exact dimension α > 0, then the associated GMC measure ν is non-degenerate and is almost surely exact dimensional with dimension α − γ 2 2 , provided γ 2 2 < α. We then show that if ν t is a Hölder-continuously parameterized family of measures then the total mass of ν t varies Hölder-continuously with t, provided that γ is sufficiently small. As an application we show that if γ < 0.28, then, almost surely, the orthogonal projections of the γ-Liouville quantum gravity measure µ on a rotund convex domain D in all directions are simultaneously absolutely continuous with respect to Lebesgue measure with Hölder continuous densities. Furthermore, µ has positive Fourier dimension almost surely.1991 Mathematics Subject Classification. 28A80, 60D05, 81T40.

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Section: 2mentioning

confidence: 99%

Exact dimensionality and projection properties of Gaussian multiplicative chaos measures

Falconer

Jin

2019

Trans. Amer. Math. Soc.

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“…For d = 1, the middle-thirds Cantor set in R has Fourier dimension 0 and Hausdorff dimension ln 2/ ln 3. Some subtle properties of Fourier dimension are studied by Ekström, Persson, and Schmeling [16] and Fraser, Orponen, and Sahlsten [19].…”

Section: 1)mentioning

confidence: 99%

Explicit Salem sets, Fourier restriction, and metric Diophantine approximation in thep-adic numbers

Fraser¹,

Hambrook²

2019

Proceedings of the Royal Society of Edinburgh: Section A Mathem

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We exhibit the first explicit examples of Salem sets in Qp of every dimension 0 < α < 1 by showing that certain sets of well-approximable p-adic numbers are Salem sets. We construct measures supported on these sets that satisfy essentially optimal Fourier decay and upper regularity conditions, and we observe that these conditions imply that the measures satisfy strong Fourier restriction inequalities. We also partially generalize our results to higher dimensions. Our results extend theorems of Kaufman, Papadimitropoulos, and Hambrook from the real to the p-adic setting.Points, spheres, and balls in R d are Salem sets of dimension 0, d − 1, and d, respectively. Salem sets are named for Raphaël Salem [37], who proved the existence of Salem sets in R of every dimension 0 < α < 1 via a construction of Cantor sets with random contraction ratios. Kahane [24] proved the existence of Salem sets in R d of every dimension 0 < α < d by considering trajectories of Brownian motion and more general stochastic processes. There are many other random constructions of Salem sets in R d (see [7], [12], [15], [26], [38]).The random constructions of Salem sets mentioned above are unsatisfactory in that they do not give explicit Salem sets. At best, they give families whose members are (with respect to some measure) almost all Salem sets.Kaufman [25] was the first to give a construction of explicit Salem sets in R of every dimension 0 < α < 1. His construction comes from number theory and is (arguably) simpler than the random constructions mentioned above.For τ ∈ R, the set of τ -well-approximable real numbers is1] : |qx − r| ≤ max(|q|, |r|) −τ for infinitely many (q, r) ∈ Z 2 . A classic application of Dirichlet's pigeonhole principle is that E(τ ) = [−1, 1] when τ ≤ 1. Jarník [22] and Besicovitch [6] proved that E(τ ) has Hausdorff dimension 2/(1 + τ ) when τ > 1. Much further work has been done on metric properties of E(τ ) and various generalizations of it. For details, we direct the reader to the recent works [2], [4], [5], and references therein. Kaufman [25] proved Theorem 1.2.1 (Kaufman). For every τ > 1, E(τ ) is a Salem set of Hausdorff and Fourier dimension 2/(1 + τ ). Moreover, there exists a Borel probability measure µ supported on E(τ ) such that | µ(ξ)| |ξ| −1/(1+τ ) ln(1 + |ξ|) ∀ξ ∈ R, ξ = 0. All known constructions of explicit Salem sets in R d of dimension α / ∈ {0, d − 1, d} are based on Kaufman's construction. Bluhm [8] and Hambrook [20] generalized Kaufman's construction to show that some sets closely related to E(τ ) are also Salem sets in R. Bluhm [8] also observed that the radial set x ∈ R d : |x| ∈ E(τ ) (here and nowhere else | · | is the Euclidean norm on R d ) is a Salem set of dimension d − 1 + 2/(1 + τ ) when τ > 1. Hambrook [21] generalized Kaufman's construction to give explicit Salem sets in R 2 of every dimension 0 < α < 2.

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“…As general references for Hausdorff dimension, Fourier dimension, and the Fourier analysis of measures, we give [24], [25], and [31]. The recent papers [10] and [11] (to name just two) also discuss aspects of the theory of Fourier dimension.…”

Section: Motivation: Explicit Salem Setsmentioning

confidence: 99%

Explicit Salem sets and applications to metrical Diophantine approximation

Hambrook

2018

Trans. Amer. Math. Soc.

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Let Q be an infinite subset of Z, let Ψ : Z → [0, ∞) be positive on Q, and let θ ∈ R. DefineWe prove a lower bound on the Fourier dimension of E(Q, Ψ, θ). This generalizes theorems of Kaufman and Bluhm and yields new explicit examples of Salem sets. We give applications to metrical Diophantine approximation, including determining the Hausdorff dimension of E(Q, Ψ, θ) in new cases. We also prove a higher-dimensional analog of our result.

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On the Fourier dimension and a modification

Cited by 23 publications

References 13 publications

Exact dimensionality and projection properties of Gaussian multiplicative chaos measures

Exact dimensionality and projection properties of Gaussian multiplicative chaos measures

Explicit Salem sets, Fourier restriction, and metric Diophantine approximation in thep-adic numbers

Explicit Salem sets and applications to metrical Diophantine approximation

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