How large dimension guarantees a given angle?

Harangi, Viktor; Keleti, Tamás; Kiss, Gergely; Maga, Péter; Máthé, András; Mattila, Pertti; Strenner, Balázs

doi:10.1007/s00605-012-0455-0

Cited by 19 publications

(20 citation statements)

References 9 publications

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“…Harangi, Keleti, Kiss, Maga, Máthé, Matilla, and Strenner [15] give upper bounds on C(n, θ) (which they show is tight for θ = 0, π), and Máthé [28] provides lower bounds. Their results are summarized below.…”

Section: Examplesmentioning

confidence: 94%

Finite configurations in sparse sets

Chan

Łaba

Pramanik

2016

JAMA

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Let E ⊆ R n be a closed set of Hausdorff dimension α. For m ≥ n, let {B 1 , . . . , B k } be n × (m − n) matrices. We prove that if the system of matrices B j is non-degenerate in a suitable sense, α is sufficiently close to n, and if E supports a probability measure obeying appropriate dimensionality and Fourier decay conditions, then for a range of m depending on n and k, the set E contains a translate of a non-trivial k-point configuration {B 1 y, . . . , B k y}. As a consequence, we are able to establish existence of certain geometric configurations in Salem sets (such as parallelograms in R n and isosceles right triangles in R 2 ). This can be viewed as a multidimensional analogue of the result of [25] on 3-term arithmetic progressions in subsets of R.

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Section: Examplesmentioning

confidence: 94%

Finite configurations in sparse sets

Chan

Łaba

Pramanik

2016

JAMA

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“…On the other hand, large Hausdorff dimensionality, while failing to ensure specific patterns, is sometimes sufficient to ensure existence or even abundance of certain configuration classes; see for instance the work of Iosevich et al [5,6,2,7]. Harangi, Keleti, Kiss, Maga, Máthé, Mattila, and Strenner in [8] show that sets of sufficiently large Hausdorff dimension contain points that generate specific angles.…”

Section: Introductionmentioning

confidence: 99%

“…To the best of our knowledge, such uniform avoidance results are new. Some points of tenuous similarity may be found in [8], where the authors construct sets that avoid angles within a specific range, but the ideas, methods and goals are very different.…”

Section: Introductionmentioning

confidence: 99%

Large sets avoiding patterns

Fraser

Pramanik

2018

Analysis & PDE

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We construct subsets of Euclidean space of large Hausdorff dimension and full Minkowski dimension that do not contain nontrivial patterns described by the zero sets of functions. The results are of two types. Given a countable collection of v-variate vector-valued functions fq : (R n ) v → R m satisfying a mild regularity condition, we obtain a subset of R n of Hausdorff dimension m v−1 that avoids the zeros of fq for every q. We also find a set that simultaneously avoids the zero sets of a family of uncountably many functions sharing the same linearization. In contrast with previous work, our construction allows for non-polynomial functions as well as uncountably many patterns. In addition, it highlights the dimensional dependence of the avoiding set on v, the number of input variables.

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“…However, there are sequences of distances (d i ), d i → 0, and compact sets E ⊂ R n of dimension n such that d i / ∈ D(E) for every i. There has been recent interest in studying the set of angles A(E) formed by three-point subsets of a given E ⊂ R n , see [5,6,7]. A particularly interesting open question is the following: What minimum dimension of a set E ⊂ R n guarantees that A(E) has positive Lebesgue measure?…”

Section: Introductionmentioning

confidence: 99%