How large dimension guarantees a given angle?
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Publication:359646
DOI10.1007/S00605-012-0455-0zbMath1279.28009arXiv1101.1426OpenAlexW2052895443MaRDI QIDQ359646
Viktor Harangi, Pertti Mattila, Gergely Kiss, Péter Maga, Balázs Strenner, András Máthé, Tamás Keleti
Publication date: 12 August 2013
Published in: Monatshefte für Mathematik (Search for Journal in Brave)
Abstract: We study the following two problems: (1) Given $nge 2$ and $al$, how large Hausdorff dimension can a compact set $AsuRn$ have if $A$ does not contain three points that form an angle $al$? (2) Given $al$ and $de$, how large Hausdorff dimension can a %compact subset $A$ of a Euclidean space have if $A$ does not contain three points that form an angle in the $de$-neighborhood of $al$? An interesting phenomenon is that different angles show different behaviour in the above problems. Apart from the clearly special extreme angles 0 and $180^circ$, the angles $60^circ,90^circ$ and $120^circ$ also play special role in problem (2): the maximal dimension is smaller for these special angles than for the other angles. In problem (1) the angle $90^circ$ seems to behave differently from other angles.
Full work available at URL: https://arxiv.org/abs/1101.1426
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Related Items (13)
Finite configurations in sparse sets ⋮ Avoiding right angles and certain Hamming distances ⋮ A Mattila-Sjölin theorem for triangles ⋮ On some properties of sparse sets: a survey ⋮ Large dimensional sets not containing a given angle ⋮ On Radii of spheres determined by subsets of Euclidean space ⋮ Occurrence of right angles in vector spaces over finite fields ⋮ Large sets avoiding patterns ⋮ Sets of large dimension not containing polynomial configurations ⋮ Maximum subsets of \(\mathbb{F}^n_q\) containing no right angles ⋮ A framework for constructing sets without configurations ⋮ Patterns in thick compact sets ⋮ Fourier dimension and avoidance of linear patterns
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