Large Sets Avoiding Patterns

TL;DR

This paper constructs large fractal sets in Euclidean space that avoid complex pattern zero sets, extending previous work to include non-polynomial functions and uncountably many patterns, with results depending on the number of variables.

Contribution

It introduces a method to build large sets avoiding zeros of broad classes of functions, including non-polynomial and uncountably many, with explicit dimensional bounds.

Findings

Constructed sets of large Hausdorff and Minkowski dimension avoiding pattern zeros.

Extended avoidance results to non-polynomial functions and uncountably many patterns.

Demonstrated the dependence of set dimension on the number of input variables.

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 $f_q : (\mathbb{R}^{n})^v \to \mathbb{R}^m$ 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.