On Fourier Analytic Properties of Graphs

Fraser, Jonathan M.; Orponen, Tuomas; Sahlsten, Tuomas

doi:10.1093/imrn/rnt015

Cited by 16 publications

(13 citation statements)

References 24 publications

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“…In 1993, Kahane [16] outlined the problem explicitly. Kahane's problem for graphs, even in the case of the standard Brownian motion W t , however, remained open for quite a while until, together with T. Orponen, we established that the Brownian graph G(W ) is almost surely not a Salem set [9]. It turned out that the reason for this is purely geometric: the proof was based on the following application of a Fourier-analytic version of Marstrand's slicing lemma.…”

Section: Introduction and Resultsmentioning

confidence: 99%

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On the Fourier analytic structure of the Brownian graph

Fraser

Sahlsten

2018

Analysis & PDE

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In a previous article (Int. Math. Res. Not. 2014:10 (2014, 2730-2745) T. Orponen and the authors proved that the Fourier dimension of the graph of any real-valued function on ‫ޒ‬ is bounded above by 1. This partially answered a question of Kahane (1993) by showing that the graph of the Wiener process W t (Brownian motion) is almost surely not a Salem set. In this article we complement this result by showing that the Fourier dimension of the graph of W t is almost surely 1. In the proof we introduce a method based on Itô calculus to estimate Fourier transforms by reformulating the question in the language of Itô drift-diffusion processes and combine it with the classical work of Kahane on Brownian images.

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Section: Introduction and Resultsmentioning

confidence: 99%

“…The methods in [9] are purely geometric and involve no stochastic properties of Brownian motion. They also do not shed any light on the precise value for the Fourier dimension of G(W ).…”

Section: Introduction and Resultsmentioning

confidence: 99%

On the Fourier analytic structure of the Brownian graph

Fraser

Sahlsten

2018

Analysis & PDE

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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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“…Fraser et al found in [FOS3] another interesting application for this inequality: they showed that any one-dimensional graph has Fourier dimension 1. More precisely, in general dimensions Theorem 8.5 For any function f : A → R n−m , A ⊂ R m , and for its graph…”

Section: Slicing Theoremsmentioning

confidence: 99%

Geometry and Analysis of Fractals

Ding¹,

Lau²

2014

Springer Proceedings in Mathematics &Amp; Statistics

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This book series features volumes composed of selected contributions from workshops and conferences in all areas of current research in mathematics and statistics, including OR and optimization. In addition to an overall evaluation of the interest, scientific quality, and timeliness of each proposal at the hands of the publisher, individual contributions are all refereed to the high quality standards of leading journals in the field. Thus, this series provides the research community with well-edited, authoritative reports on developments in the most exciting areas of mathematical and statistical research today.More information about this series at

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On Fourier Analytic Properties of Graphs

Cited by 16 publications

References 24 publications

On the Fourier analytic structure of the Brownian graph

On the Fourier analytic structure of the Brownian graph

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

Geometry and Analysis of Fractals

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