Cubes and their centers

Thornton, Riley

doi:10.1007/s10474-017-0729-z

Cited by 14 publications

(29 citation statements)

References 10 publications

(18 reference statements)

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“…In particular, they proved that sets of fractional dimension can differentiate some L p spaces. A similar geometric problem, in which circles are replaced by k-skeletons of ncubes with axes-parallel sides was recently studied by T. Keleti, D. Nagy and the second author in [4], for the case n = 2, and by R. Thornton in [9] for n ≥ 3. In [4] it was shown that a set in the plane containing a 1-skeleton with center in every point of [0, 1] 2 can have Lebesgue measure 0, and it was investigated how small its fractal dimension can be for different notions of dimension.…”

Section: Introduction and Main Resultsmentioning

confidence: 97%

“…In [4] it was shown that a set in the plane containing a 1-skeleton with center in every point of [0, 1] 2 can have Lebesgue measure 0, and it was investigated how small its fractal dimension can be for different notions of dimension. The arguments from [4,9] are direct and do not involve any maximal operators. The goal of this paper is to study a natural k-skeleton maximal operator associated to this geometric problem.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

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Maximal operators for cube skeletons

Olivo¹,

Shmerkin²

2020

Ann. Acad. Sci. Fenn. Math.

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We study discretized maximal operators associated to averaging over (neighborhoods of) squares in the plane and, more generally, k-skeletons in R n . Although these operators are known not to be bounded on any L p , we obtain nearly sharp L p bounds for every small discretization scale. These results are motivated by, and partially extend, recent results of T. Keleti, D. Nagy and P. Shmerkin, and of R. Thornton, on sets that contain a scaled k-sekeleton of the unit cube with center in every point of R n .

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Section: Introduction and Main Resultsmentioning

confidence: 97%

Section: Introduction and Main Resultsmentioning

confidence: 99%

Maximal operators for cube skeletons

Olivo¹,

Shmerkin²

2020

Ann. Acad. Sci. Fenn. Math.

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“…Roughly speaking, the k-dimensional boundary of a cube with axes parallel sides in R n for 0 ≤ k < n. In R 3 , for example, the case k = 2 consists in considering the faces of the cube, k = 1 is for the set of edges of the cube and the case k = 0 describes the vertices. The problem of finding minimal values of the size for sets in R n containing k-skeletons centered at any point was recently studied by Keleti, Nagy and Shmerkin in [KNS18] and also by Thornton in [Tho17]. The main result in [KNS18] is that there exists a set containing a 1-skeleton centered at any point of the unit square [0, 1] 2 having Lebesgue measure zero and, moreover, with zero Hausdorff dimension.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Weighted Estimates for Maximal Functions Associated to Skeletons

Olivo

Rela

2019

J Geom Anal

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We provide quantitative weighted estimates for the L p (w) norm of a maximal operator associated to cube skeletons in R n . The method of proof differs from the usual in the area of weighted inequalities since there are no covering arguments suitable for the geometry of skeletons. We use instead a combinatorial strategy that allows to obtain, after a linearization and discretization, L p bounds for the maximal operator from an estimate related to intersections between skeletons and k-planes.1991 Mathematics Subject Classification. Primary: 42B25. Secondary: 43A85.

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“…In Section 5 we study the case when T is the k-skeleton (0 ≤ k < n) of a fixed axis parallel n-dimensional cube centered at the origin or the k-skeleton of a more general polytope. For the k-skeleton of axis parallel cubes Thornton [22] generalized the above mentioned two-dimensional results for packing and box dimensions (Theorem 5.2), found the estimate dim H B ≥ max(k, dim H S − 1) for Hausdorff dimension and posed the conjecture the this estimate is sharp. This conjecture was proved by Chang, Csörnyei, Héra and the author [3] not only for cubes but for more general polytopes (Theorem 5.6).…”

Section: Introductionmentioning

confidence: 89%

“…[11], [22]) If 0 ≤ k < n and B ⊂ R n contains a k-dimensional skeleton of an n-dimensional axis-parallel cube centered at every point S ⊂ R n of dimension s (for some dimension) then the best lower bound for the dimension (for the same dimension) of B is shown in the last column of the following table.…”

Section: Theorem 52 (Thornton [22])mentioning

confidence: 99%

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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Cubes and their centers

Cited by 14 publications

References 10 publications

Maximal operators for cube skeletons

Maximal operators for cube skeletons

Weighted Estimates for Maximal Functions Associated to Skeletons

Small Union with Large Set of Centers

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