# A Mattila-Sjölin theorem for triangles

TL;DR: For a compact set E⊂Rd, d ≥ 4, it was shown in this paper that if the Hausdorff dimension of E is larger than 23d+1, then the set of congruence classes of triangles formed by triples of points of E has non-empty interior.

About: This article is published in Journal of Functional Analysis.The article was published on 2022-12-01 and is currently open access. It has received 1 citations till now. The article focuses on the topics: Mathematics & Mathematics.

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29 May 2023

TL;DR: In this paper , restricted Falconer distance problems are introduced, which lie in between the classical version and its pinned variant, and they are shown to have non-empty interior if the diagonal distance set has a nonempty interior.

Abstract: We introduce a new class of Falconer distance problems, which we call restricted Falconer distance problems, that lie in between the classical version and its pinned variant. A particular model we study is the diagonal distance set $$\Delta^{diag}(E)= \{ \,|(x,x)-(y_1,y_2)| \, :\, x,\,y_1,\,y_2\, \in E\, \}$$ which we show has non-empty interior if $\dim(E)>\frac{2d+1}{3}$. Standard pinned variants of the Falconer distance problem either can not guarantee a pin on the diagonal or yield worse dimensional thresholds. A key tool for our result is an $L^p$ improving estimate for the bilinear spherical averaging operator with decay on frequency scales.

1 citations

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01 Jan 2009

TL;DR: In this paper, the Carleson-Hunt Theorem is used to describe the smoothness and function spaces of non-convolutional non-convolutional types.

Abstract: Preface.- Smoothness and Function Spaces.- BMO and Carleson Measures.- Singular Integrals of Nonconvolution Type.- Weighted Inequalities.- Boundedness and Convergence of Fourier Integrals.- Time-Frequency Analysis and the Carleson-Hunt Theorem.- Multilinear Harmonic Analysis.- Glossary.- References.- Index.

1,195 citations

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235 citations

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TL;DR: The best known result on distance sets is due to Steinhaus [11], namely, if E ⊂ ℝn is measurable with positive n-dimensional Lebesgue measure, then D(E) contains an interval [0, e] for some e > 0 as mentioned in this paper.

Abstract: If E is a subset of ℝn (n ≥ 1) we define the distance set of E asThe best known result on distance sets is due to Steinhaus [11], namely, that, if E ⊂ ℝn is measurable with positive n-dimensional Lebesgue measure, then D(E) contains an interval [0, e) for some e > 0. A number of variations of this have been examined, see Falconer [6, p. 108] and the references cited therein.

225 citations

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17 Sep 2003

TL;DR: The Schwartz space Fourier inversion and the Plancherel theorem have been studied in this article, and the stationary phase method has been applied to the Kakeya problem.

Abstract: The $L^1$ Fourier transform The Schwartz space Fourier inversion and the Plancherel theorem Some specifics, and $L^p$ for $p<2$ The uncertainty principle The stationary phase method The restriction problem Hausdorff measures Sets with maximal Fourier dimension and distance sets The Kakeya problem Recent work connected with the Kakeya problem Bibliography for Chapter 11 Historical notes Bibliography.

187 citations

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TL;DR: In this article, the authors studied the Erdos/Falconer distance problem in vector spaces over finite fields and developed a Fourier analytic machinery, analogous to that developed by Mattila in the continuous case, for the study of distance sets in order to provide estimates for minimum cardinality of the distance set.

Abstract: We study the Erdos/Falconer distance problem in vector spaces over finite fields. Let be a finite field with elements and take , . We develop a Fourier analytic machinery, analogous to that developed by Mattila in the continuous case, for the study of distance sets in to provide estimates for minimum cardinality of the distance set in terms of the cardinality of . Bounds for Gauss and Kloosterman sums play an important role in the proof.

182 citations