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Sets of large dimension not containing polynomial configurations
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Máthé, András (2017) Sets of large dimension not containing polynomial configurations. Advances in Mathematics, 316 . pp. 691-709. doi:10.1016/j.aim.2017.01.002 ISSN 0001-8708.
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Official URL: http://dx.doi.org/10.1016/j.aim.2017.01.002
Abstract
The main result of this paper is the following. Given countably many multivariate polynomials with rational coefficients and maximum degree d, we construct a compact set of Hausdorff dimension which does not contain finite point configurations corresponding to the zero sets of the given polynomials.
Given a set , we study the angles determined by three-point subsets of E. The main result implies the existence of a compact set in of Hausdorff dimension which does not contain the angle . (This is known to be sharp if n is even.) We show that there is a compact set of Hausdorff dimension which does not contain an angle in any given countable set. We also construct a compact set of Hausdorff dimension for which the set of angles determined by E is Lebesgue null.
In the other direction, we present a result that every set of sufficiently large dimension contains an angle ε close to any given angle.
The main result can also be applied to distance sets. As a corollary we obtain a compact set () of Hausdorff dimension which does not contain rational distances nor collinear points, for which the distance set is Lebesgue null, moreover, every distance and direction is realised only at most once by E.
Item Type: | Journal Article | ||||||||
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Divisions: | Faculty of Science, Engineering and Medicine > Science > Mathematics | ||||||||
Journal or Publication Title: | Advances in Mathematics | ||||||||
Publisher: | Academic Press | ||||||||
ISSN: | 0001-8708 | ||||||||
Official Date: | 20 August 2017 | ||||||||
Dates: |
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Volume: | 316 | ||||||||
Page Range: | pp. 691-709 | ||||||||
DOI: | 10.1016/j.aim.2017.01.002 | ||||||||
Status: | Peer Reviewed | ||||||||
Publication Status: | Published | ||||||||
Access rights to Published version: | Restricted or Subscription Access | ||||||||
Date of first compliant deposit: | 19 November 2019 | ||||||||
Date of first compliant Open Access: | 19 November 2019 | ||||||||
Open Access Version: |
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