Large sets avoiding patterns

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 $f_{q} : {(ℝ^{n})}^{v} \to ℝ^{m}$ satisfying a mild regularity condition, we obtain a subset of $ℝ^{n}$ of Hausdorff dimension $m ∕ (v - 1)$ that avoids the zeros of $f_{q}$ 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 nonpolynomial 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.

Keywords

geometric measure theory, configurations, Hausdorff dimension, Minkowski dimension

Mathematical Subject Classification 2010

Primary: 28A78, 28A80, 26B10, 05B30

Milestones

Received: 9 September 2016

Revised: 8 June 2017

Accepted: 2 January 2018

Published: 11 April 2018

Authors

Robert Fraser
University of British Columbia Vancouver, BC Canada

Malabika Pramanik
University of British Columbia Vancouver, BC Canada

Vol. 11, No. 5, 2018

Robert Fraser and Malabika Pramanik

Abstract

Keywords

Mathematical Subject Classification 2010

Milestones

Authors