Small unions of affine subspaces and skeletons via Baire category

Chang, Alan; Csörnyei, Marianna; Héra, Kornélia; Keleti, Tamás

doi:10.1016/j.aim.2018.02.009

Cited by 14 publications

(44 citation statements)

References 19 publications

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“…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). To obtain this result first the smallest Hausdorff dimension of B was determined for any fixed compact S, in other words, instead of dim H S we fixed S itself (Theorem 5.9).…”

Section: Introductionmentioning

confidence: 88%

“…If 0 is contained in one of the k-dimensional affine subspaces defined by T then the projection argument implies dim H B ≥ max(dim H S, k) and it is also proved in [3] that in this case this is sharp. So the minimal Hausdorff dimension of a compact B such that (♠) holds for some compact S with dim H S = s is max{s, k} if 0 is contained in one of the k-dimensional affine subspaces defined by T and max{s − 1, k} otherwise.…”

Section: Theorem 52 (Thornton [22])mentioning

confidence: 95%

“…This conjecture was proved by A. Chang, M. Csörnyei, K. Héra and the author [3], even for more general polytopes: Theorem 5.6. ( [3]) Let 0 ≤ k < n and T be the k-skeleton of an n-dimensional polytope such that 0 is not contained in any of the k-dimensional affine subspaces defined by T . Then for every s ∈ [0, n] there exist compact sets S and B such that (♠) (∀x ∈ S) (∃r > 0) x + rT ⊂ B, dim H S = s and dim H B = max(s − 1, k).…”

Section: Theorem 52 (Thornton [22])mentioning

confidence: 97%

“…It turns out that this k + 1 is sharp, B must have Hausdorff dimension at least k + 1 (Corollary 6.4). Most of the results of this section are very recent results of Chang, Csörnyei, Héra and the author [3], the last mentioned lower estimate for dim H B is a very recent result of Héra, Máthé and the author [8].…”

Section: Introductionmentioning

confidence: 98%

“…To get the above mentioned constructions of Chang, Csörnyei, Héra and the author [3] a slightly different problem was studied first: instead of fixing the Hausdorff dimension s of S, the set S itself was fixed.…”

Section: Theorem 52 (Thornton [22])mentioning

confidence: 99%

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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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Section: Introductionmentioning

confidence: 88%

Section: Theorem 52 (Thornton [22])mentioning

confidence: 95%

Section: Theorem 52 (Thornton [22])mentioning

confidence: 97%

Section: Introductionmentioning

confidence: 98%

Section: Theorem 52 (Thornton [22])mentioning

confidence: 99%

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Small Union with Large Set of Centers

Keleti

2017

Trends in Mathematics

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A Strong-Type Furstenberg–Sárközy Theorem for Sets of Positive Measure

2023

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For every $$\beta \in (0,\infty )$$ β ∈ ( 0 , ∞ ) , $$\beta \ne 1$$ β ≠ 1 , we prove that a positive measure subset A of the unit square contains a point $$(x_0,y_0)$$ ( x 0 , y 0 ) such that A nontrivially intersects curves $$y-y_0 = a (x-x_0)^\beta $$ y - y 0 = a ( x - x 0 ) β for a whole interval $$I\subseteq (0,\infty )$$ I ⊆ ( 0 , ∞ ) of parameters $$a\in I$$ a ∈ I . A classical Nikodym set counterexample prevents one to take $$\beta =1$$ β = 1 , which is the case of straight lines. Moreover, for a planar set A of positive density, we show that the interval I can be arbitrarily large on the logarithmic scale. These results can be thought of as Bourgain-style large-set variants of a recent continuous-parameter Sárközy-type theorem by Kuca, Orponen, and Sahlsten.

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Cube Packings in Euclidean Spaces

2021

Mathematika

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In this paper, we study some cube packing problems. In particular, we are interested in compact subsets of double-struckRn,n⩾2, which contain boundaries of cubes with all side lengths in (0,1). We show here that such sets must have lower box dimension at least n−0.5, and we will also provide sharp examples. We also show here that such sets must be large in general in a precise sense which is also introduced in this paper.

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Small unions of affine subspaces and skeletons via Baire category

Cited by 14 publications

References 19 publications

Small Union with Large Set of Centers

Small Union with Large Set of Centers

A Strong-Type Furstenberg–Sárközy Theorem for Sets of Positive Measure

Cube Packings in Euclidean Spaces

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