Pinned distance problem, slicing measures, and local smoothing estimates

Iosevich, Alex; Liu, Bochen

doi:10.1090/tran/7693

Cited by 14 publications

(17 citation statements)

References 22 publications

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“…Y is a smooth surface of codimension k. As in the paper [10] by Grafakos, Palsson and the first two authors, for a compact subset E ⊂ R d , we define the (two-point) Φ-configuration set of E as Our goal is to find a threshold, s 0 = s 0 (Φ), such that if dim H (E) > s 0 then ∆ Φ (E) has nonempty interior. Similar questions have been studied in, or are accessible to the methods of, a number of works, e.g., [11,2,5,23,24,21]. (There is of course an extensive literature on the related Falconer distance problem [8], and its generalizations to configurations, where the question is what lower bound on dim H (E) ensures that ∆ Φ (E) has positive Lebesque measure [41,7,6,15].)…”

Section: Introductionmentioning

confidence: 96%

Configuration Sets with Nonempty Interior

Greenleaf¹,

Iosevich²,

Taylor³

2019

J Geom Anal

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A theorem of Steinhaus states that if E ⊂ R d has positive Lebesgue measure, then the difference set E − E contains a neighborhood of 0. Similarly, if E merely has Hausdorff dimension dim H (E) > (d + 1)/2, a result of Mattila and Sjölin states that the distance set ∆(E) ⊂ R contains an open interval. In this work, we study such results from a general viewpoint, replacing E − E or ∆(E) with more general Φ -configurations for a class of Φ : R d × R d → R k , and showing that, under suitable lower bounds on dim H (E) and a regularity assumption on the family of generalized Radon transforms associated with Φ, it follows that the set ∆ Φ (E) of Φ-configurations in E has nonempty interior in R k . Further extensions hold for Φ -configurations generated by two sets, E and F , in spaces of possibly different dimensions and with suitable lower bounds on dim H (E) +

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Section: Introductionmentioning

confidence: 96%

Configuration Sets with Nonempty Interior

Greenleaf¹,

Iosevich²,

Taylor³

2019

J Geom Anal

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“…Peres and Schlag ( [32]) proved that if E Ă R d , d ě 2 and dim H pEq ą d` 1 2 , then Lp∆ x pEqq ą 0 for every x P E except for a set of small Hausdorff dimension. Improvements on the size of the exceptional set were obtained by the second listed author and Liu in [20].…”

Section: Introductionmentioning

confidence: 99%

On Falconer’s distance set problem in the plane

et al. 2019

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If E Ă R 2 is a compact set of Hausdorff dimension greater than 5{4, we prove that there is a point x P E so that the set of distances t|x´y|u yPE has positive Lebesgue measure. 1 arXiv:1808.09346v1 [math.CA] 28 Aug 2018 the complications of replacing the upper Minkowski dimension by the Hausdorff dimension in the claim above.Falconer's distance problem can be thought of as a continuous analogue of a combinatorial problem raised by Erdős in [11]: given a set P of N points in R d , what is the smallest possible cardinality of ∆pP q. A grid is the best known example in all dimensions. In two dimensions, Guth and Katz [15] proved a lower bound for |∆pP q| which nearly matches the grid example (up to a factor of log 1{2 N ). In higher dimensions, there is a larger gap, and the best known result is due to Solymosi and Vu [35]. The Erdős distinct distance problem also makes sense for general norms and much less is known about it. In the planar case, if K is smooth and has strictly positive curvature, the best known bound says that if |P | " N , then |∆ K pP q| Á N 3{4 , with stronger estimates established by Garibaldi in special cases ([13]). There is a conversion mechanism to go from Falconer-type results to Erdőstype results that was developed by the second author together with Hoffman ([17]), Laba ([19]), and Rudnev and Uriarte-Tuero ([18]). It gives estimates for point sets that are fairly spread out. Applying the conversion mechanism to Theorem 1.3 we get the following corollary:Corollary 1.5. Let K be a symmetric convex body in R 2 whose boundary BK is C 8 smooth and has strictly positive curvature. Let P be a set of N points in r0, 1s 2 so that the distance between any two points is Á N´1 {2 . Then there exists x P P such that (1.1) |∆ K,x pP q| Ç N 4 5 .

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“…Both results imply that if dim H (A) > 3/2, then there is y ∈ A such that dim H (∆ y A) = 1, and it is unknown whether 3/2 can be replaced by a smaller number. We remark that the results of both [24] and [12] extend to higher dimensions. These were the best known results towards Falconer's problem in the plane for general sets prior to this article.…”

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confidence: 59%

“…Recently, A. Iosevich and B. Liu [12] proved that (1.3) remains true with 3 + 3t − 3s in the right-hand side. This is an improvement in some parts of the parameter region.…”

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confidence: 99%

New Bounds on the Dimensions of Planar Distance Sets

Keleti

Shmerkin

2019

Geom. Funct. Anal.

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We prove new bounds on the dimensions of distance sets and pinned distance sets of planar sets. Among other results, we show that if A ⊂ R 2 is a Borel set of Hausdorff dimension s > 1, then its distance set has Hausdorff dimension at least 37/54 ≈ 0.685. Moreover, if s ∈ (1, 3/2], then outside of a set of exceptional y of Hausdorff dimension at most 1, the pinned distance set {|x − y| : x ∈ A} has Hausdorff dimension ≥ 2 3 s and packing dimension at least 1 4 (1 + s + 3s(2 − s)) ≥ 0.933. These estimates improve upon the existing ones by Bourgain, Wolff, Peres-Schlag and Iosevich-Liu for sets of Hausdorff dimension > 1. Our proof uses a multiscale decomposition of measures in which, unlike previous works, we are able to choose the scales subject to certain constrains. This leads to a combinatorial problem, which is a key new ingredient of our approach, and which we solve completely by optimizing certain variation of Lipschitz functions.2010 Mathematics Subject Classification. Primary: 28A75, 28A80; Secondary: 26A16, 49Q15.

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Pinned distance problem, slicing measures, and local smoothing estimates

Abstract: We improve the Peres-Schlag result on pinned distances in sets of a given Hausdorff dimension. In particular, for Euclidean distances, with

Cited by 14 publications

References 22 publications

Configuration Sets with Nonempty Interior

Configuration Sets with Nonempty Interior

On Falconer’s distance set problem in the plane

New Bounds on the Dimensions of Planar Distance Sets

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