Squares and their centers

Keleti, Tamás; Nagy, Dániel T.; Shmerkin, Pablo

doi:10.1007/s11854-018-0021-3

Cited by 20 publications

(68 citation statements)

References 10 publications

(17 reference statements)

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“…Most of the discrete results follow as corollaries from two lemmas: a construction based on iary expansions and a bound relating sets in dimension n and ℓ for any ℓ < n. We give the construction first. The lemma below generalizes [6,Lemma 4.3].…”

Section: Discrete Resultsmentioning

confidence: 78%

“…The main results of the paper are the generalizations of the bounds for box-counting and packing dimensions in [6]: Theorem 1.1. For any 0 ≤ k < n and any sets B, S ⊆ R n such that B contains the k-skeleton of a cube around every point in S,…”

Section: Resultsmentioning

confidence: 99%

“…The analagous discrete problem of optimizing cardinality with respect to some combinatorial constraint is the domain of extremal combinatorics. This paper and [6] illuminate some relationships between these two areas of research.…”

Section: Notion Of Sizementioning

confidence: 92%

“…In [6] the authors find sharp bounds for the Hausdorff, boxcounting, and packing dimensions of sets S, B ⊆ R 2 where B contains either the vertices or boundary of an axes-parallel square around every point in S, and cardinality bounds for finite sets satisfying discrete versions of these conditions. Their results are summarized in the following table.…”

Section: 1mentioning

confidence: 99%

“…The authors in [6] were inspired by Bourgain [2] and Marstrand's [9] results about packing circles, which in turn completed work done by E. Stein work on n-spheres for n ≥ 3 [13] The problem we study, the original problem in [6], and the problems settled by Stein and (independently) Bourgain and Marstrand are members of the abundant family of "Kakeya type" problems. The problem of minimizing size, in the sense of some measure of fractal dimension, over some family of sets in R d occurs commonly in geometric measure theory and other areas of analysis.…”

Section: Notion Of Sizementioning

confidence: 99%

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Cubes and their centers

Thornton

2017

Acta Math. Hungar.

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Abstract. We study the relationship between the sizes of sets B, S in R n where B contains the k-skeleton of an axes-parallel cube around each point in S, generalizing the results of Keleti, Nagy, and Shmerkin [6] about such sets in the plane. We find sharp estimates for the possible packing and boxcounting dimensions for B and S. These estimates follow from related cardinality bounds for sets containing the discrete skeleta of cubes around a finite set of a given size. The Katona-Kruskal Theorem from hypergraph theory plays an important role. We also find partial results for the Haussdorff dimension and settle an analogous question for the dual polytope of the cube, the orthoplex. Introduction and Statements of Results 1.1.Introduction. In [6] the authors find sharp bounds for the Hausdorff, boxcounting, and packing dimensions of sets S, B ⊆ R 2 where B contains either the vertices or boundary of an axes-parallel square around every point in S, and cardinality bounds for finite sets satisfying discrete versions of these conditions. Their results are summarized in the following table. If S has size s for the given notion of size, then a sharp lower bound for the size of B is given in terms of s: Notion of SizeVertex problem Boundary problem (0-skeleton of a 2-cube) (1-skeleton of a 2-cube)

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Section: Discrete Resultsmentioning

confidence: 78%

Section: Resultsmentioning

confidence: 99%

Section: Notion Of Sizementioning

confidence: 92%

Section: 1mentioning

confidence: 99%

Section: Notion Of Sizementioning

confidence: 99%

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Cubes and their centers

Thornton

2017

Acta Math. Hungar.

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Small Union with Large Set of Centers

Keleti

2017

Trends in Mathematics

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Abstract. Let T ⊂ R n be a fixed set. By a scaled copy of T around x ∈ R n we mean a set of the form x + rT for some r > 0.In this survey paper we study results about the following type of problems: How small can a set be if it contains a scaled copy of T around every point of a set of given size? We will consider the cases when T is circle or sphere centered at the origin, Cantor set in R, the boundary of a square centered at the origin, or more generally the k-skeleton (0 ≤ k < n) of an n-dimensional cube centered at the origin or the k-skeleton of a more general polytope of R n .We also study the case when we allow not only scaled copies but also scaled and rotated copies and also the case when we allow only rotated copies. IntroductionIn this survey paper we study the following type of problems. How small can a set be if it contains a scaled copy of a given set around a large set of points of R n ? More precisely:n be a fixed set and let S, B ⊂ R n be sets such that for every x ∈ S there exists an r > 0 such that x + rT ⊂ B, in other words B contains a scaled copy of T around every point of S. How small can B be if we know the size of S?In Section 2 we study the case when T is a circle or sphere centered at the origin and we present the classical deep results of Stein [19] In Section 3 we study the case when n = 1 and T is a Cantor set with 0 ∈ T . In this case there are four results: Laba and Pramanik [13] constructed Cantor sets T ⊂ [1, 2] for which the Lebesgue measure of B must be positive whenever S has positive Lebesgue measure; Máthé [16] constructed Cantor sets T for which this is false; Hochman [9] proved that if dim H S > 0 then dim H B > dim H C + δ for any porous Cantor set T , where δ > 0 depends only on dim H T and dim H S; and Máthé noticed that it is a consequence of a recent projection theorem of Bourgain [2] . In Section 4 we present the results of Nagy, Shmerkin and the author [11] about the case when n = 2 and T is the boundary or the set of vertices of a fixed axis parallel square centered at the origin. It turns out that in these cases B can be much smaller than S. If S = R 2 and T is the boundary of the square then the minimal Hausdorff dimension of B is 1 (Proposition 4.3), the minimal upper box, lower box and packing dimensions of B are all 7/4 (Theorem 4.5). If S = R 2 and T is the set 2010 Mathematics Subject Classification. 28A78.

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Measures, annuli and dimensions

Buczolich

Seuret

2023

Math. Z.

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Given a Radon probability measure $$\mu $$ μ supported in $${\mathbb {R}}^d$$ R d , we are interested in those points x around which the measure is concentrated infinitely many times on thin annuli centered at x. Depending on the lower and upper dimension of $$\mu $$ μ , the metric used in the space and the thinness of the annuli, we obtain results and examples when such points are of $$\mu $$ μ -measure 0 or of $$\mu $$ μ -measure 1. The measure concentration we study is related to “bad points” for the Poincaré recurrence theorem and to the first return times to shrinking balls under iteration generated by a weakly Markov dynamical system. The study of thin annuli and spherical averages is also important in many dimension-related problems, including Kakeya-type problems and Falconer’s distance set conjecture.

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Squares and their centers

Cited by 20 publications

References 10 publications

Cubes and their centers

Cubes and their centers

Small Union with Large Set of Centers

Measures, annuli and dimensions

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