Large Sets Avoiding Patterns

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.