Large sets avoiding patterns
From MaRDI portal
Publication:1747202
DOI10.2140/APDE.2018.11.1083zbMath1388.28006arXiv1609.03105OpenAlexW2519183108WikidataQ129976579 ScholiaQ129976579MaRDI QIDQ1747202
Malabika Pramanik, Robert Fraser
Publication date: 4 May 2018
Published in: Analysis \& PDE (Search for Journal in Brave)
Abstract: 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 $f_q : (mathbb{R}^{n})^v o mathbb{R}^m$ satisfying a mild regularity condition, we obtain a subset of $mathbb{R}^n$ of Hausdorff dimension $frac{m}{v-1}$ that avoids the zeros of $f_q$ 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.
Full work available at URL: https://arxiv.org/abs/1609.03105
Other designs, configurations (05B30) Fractals (28A80) Implicit function theorems, Jacobians, transformations with several variables (26B10) Hausdorff and packing measures (28A78)
Cites Work
- Unnamed Item
- Finite configurations in sparse sets
- Finite chains inside thin subsets of \(\mathbb{R}^d\)
- How large dimension guarantees a given angle?
- Full dimensional sets without given patterns
- On polynomial configurations in fractal sets
- Fourier transforms of measures and algebraic relations on their supports
- A group-theoretic viewpoint on Erdös-Falconer problems and the Mattila integral
- Arithmetic progressions in sets of fractional dimension
- On necklaces inside thin subsets of \(\mathbb{R}^d\)
- On triangles determined by subsets of the Euclidean plane, the associated bilinear operators and applications to discrete geometry
- Construction of one-dimensional subsets of the reals not containing similar copies of given patterns
- On sets of integers containing k elements in arithmetic progression
- On Sets of Integers Which Contain No Three Terms in Arithmetical Progression
Related Items (13)
Density theorems for anisotropic point configurations ⋮ Large subsets of Euclidean space avoiding infinite arithmetic progressions ⋮ On some properties of sparse sets: a survey ⋮ Large sets avoiding Infinite arithmetic / geometric progressions ⋮ Simplices in thin subsets of Euclidean spaces ⋮ Unnamed Item ⋮ Large sets containing no copies of a given infinite sequence ⋮ On sets containing an affine copy of bounded decreasing sequences ⋮ A framework for constructing sets without configurations ⋮ Small sets containing any pattern ⋮ Large Sets Avoiding Rough Patterns ⋮ Large sets avoiding linear patterns ⋮ Fourier dimension and avoidance of linear patterns
This page was built for publication: Large sets avoiding patterns
