# A Mattila-Sj\"olin theorem for simplices in low dimensions

12 Aug 2022-

TL;DR: In this paper , it was shown that if a compact set E ⊂ R d , d ≥ 3 , has Hausdorﬀ dimension greater than (8 k − 9) (8k − 8) d + k 8 k − 8 , then the set of simplices with vertices in E has nonempty interior.

Abstract: . In this paper we show that if a compact set E ⊂ R d , d ≥ 3 , has Hausdorﬀ dimension greater than (8 k − 9) (8 k − 8) d + k 8 k − 8 , then the set of congruence class of simplices with vertices in E has nonempty interior. By set of congruence class of simplices with vertices in E we mean where 2 ≤ k < d . This result improves our previous work [28] in the sense that we now can obtain a Hausdorﬀ dimension threshold which allow us to guarantee that the set of congruence class of triangles formed by triples of points of E has nonempty interior when d = 3 as well as extending to all simplices. The present work can be thought of as an extension of the Mattila-Sjölin theorem which establishes a non-empty interior for the distance set instead of the set of congruence classes of simplices.

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03 Oct 2022

TL;DR: In this article , the authors generalize the result of McDonald and Taylor to compact sets in R d and show that the edge lengths in C 1 × C 2 corresponding to any pinned finite tree configuration have non-empty interior.

Abstract: A BSTRACT . We generalize a result of McDonald and Taylor which concerns the size of the tuples of edge lengths in the set C 1 × C 2 utilizing the notion of thickness. Speciﬁcally, we show that C 1 , C 2 ⊂ R d compact sets with thickness satisfying τ ( C 1 ) τ ( C 2 ) > 1 , then the edge lengths in C 1 × C 2 corresponding to any pinned ﬁnite tree conﬁguration has non-empty interior. Originally proven for Cantor sets on the real line by McDonald and Taylor, we use the notion of thickness introduced by Falconer and Yavicoli which allows us to generalize the result of McDonald and Taylor to compact sets in R d .

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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.

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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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22 Jul 2015

TL;DR: In this article, the authors present a list of basic notation for measure theoretic preliminaries, including measures and intersections with planes, Bernoulli convolutions, Riesz products, and Besicovitch sets.

Abstract: Preface Acknowledgements 1. Introduction 2. Measure theoretic preliminaries 3. Fourier transforms 4. Hausdorff dimension of projections and distance sets 5. Exceptional projections and Sobolev dimension 6. Slices of measures and intersections with planes 7. Intersections of general sets and measures 8. Cantor measures 9. Bernoulli convolutions 10. Projections of the four-corner Cantor set 11. Besicovitch sets 12. Brownian motion 13. Riesz products 14. Oscillatory integrals (stationary phase) and surface measures 15. Spherical averages and distance sets 16. Proof of the Wolff-Erdogan Theorem 17. Sobolev spaces, Schrodinger equation and spherical averages 18. Generalized projections of Peres and Schlag 19. Restriction problems 20. Stationary phase and restriction 21. Fourier multipliers 22. Kakeya problems 23. Dimension of Besicovitch sets and Kakeya maximal inequalities 24. (n, k) Besicovitch sets 25. Bilinear restriction References List of basic notation Author index Subject index.

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193 citations

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TL;DR: In this paper, the authors studied the behavior of the Fourier transform of the measure p. The main result in that direction is the following: ESTIMATE OF FOURIER TRANSFORMS.

Abstract: This is related to the behavior at oo of the Fourier transform of the measure p. Our main result in that direction is the following: ESTIMATE OF FOURIER TRANSFORMS. Let 5 be a sufficiently smooth compact w-surface (possibly with boundary) embedded in R, JJL a sufficiently smooth mass distribution on S vanishing near the boundary of 5. Suppose that at each point of 5, k of the n principal curvatures are different from zero. Then

174 citations