On the distance sets of Ahlfors–David regular sets

Orponen, Tuomas

doi:10.1016/j.aim.2016.11.035

Cited by 34 publications

(43 citation statements)

References 12 publications

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“…In this section, we consider a distinct distances type problem for δ-balls in R 2 , which is related to the Falconer distance problem in R 2 . As we mentioned in the introduction, Orponen [16] and Keleti-Shmerkin [23][24] [14] essentially solved the Falconer distance problem for sets that are close to Ahlfors-David regular. Here we consider the opposite type of set -Ahlfors-David regular sets of a given dimension are packed as tightly as possible, and we consider here sets that are as spread out as possible.…”

Section: An Application To the Falconer Problemmentioning

confidence: 99%

“…Theorem 1.4 proves this conjecture up a factor of δ ǫ for sets E that are as widely spaced as possible. In the other direction, there has been some remarkable work by Orponen [16] and Keleti-Shmerkin [14] on the case when E is tightly spaced. We say that E is an Ahlfors-David regular set of δ-balls if, for each ball of E, the concentric Sδ ball contains ≈ S α balls of E. Orponen's paper [16] implies that this conjecture holds up to a factor of δ ǫ for Ahlfors-David regular sets.…”

Section: Introductionmentioning

confidence: 99%

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Incidence Estimates for Well Spaced Tubes

2019

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Section: An Application To the Falconer Problemmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Incidence Estimates for Well Spaced Tubes

2019

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“…Returning to the planar case, there have been a number of important recent results. Orponen [30] proved that if E is a compact Ahlfors-David regular set of dimension s ě 1, then ∆pEq has packing dimension 1. Note that packing dimension 1 is only slightly weaker than positive measure.…”

Section: Introductionmentioning

confidence: 99%

“…The value of δ could be made explicit but it would be very small, and so the .685... is quite striking. We will use one of the key ideas of [30] and [25] in the proof of Theorem 1.1.…”

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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“…This verifies Falconer's conjecture for this type of sets, outside of the endpoint. We remark that T. Orponen [22] and the second author [25] had previously proved weaker results of the same kind. See also [15,11] for other results on the distance sets of special classes of sets.…”

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confidence: 64%

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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On the distance sets of Ahlfors–David regular sets

Cited by 34 publications

References 12 publications

Incidence Estimates for Well Spaced Tubes

Incidence Estimates for Well Spaced Tubes

On Falconer’s distance set problem in the plane

New Bounds on the Dimensions of Planar Distance Sets

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