2016/09/11 by Robert Fraser, Malabika Pramanik
Mathematics · #Combinatorics #Countable set #Dimension (graph theory) #Discrete mathematics #Euclidean space #Hausdorff dimension #Limits and Structures in Graph Theory #Mathematical Approximation and Integration #Mathematical Dynamics and Fractals #Mathematics #Minkowski space #Uncountable set #Zero (linguistics) #Zero set #math.CA #msc:28A78
paper · pdf · doi:10.2140/apde.2018.11.1083
published as Analysis & PDE 11 (2018) 1083-1111 · 26 Pages
arxiv created 2016/09/11 · openalex publication_date 2018/04/11 · arxiv updated 2018/04/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
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 [math] -variate vector-valued functions [math] satisfying a mild regularity condition, we obtain a subset of [math] of Hausdorff dimension [math] that avoids the zeros of [math] for every [math] . 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 nonpolynomial functions, as well as uncountably many patterns. In addition, it highlights the dimensional dependence of the avoiding set on [math] , the number of input variables.