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
-variate vector-valued
functions
satisfying a mild regularity condition, we obtain a subset of
of Hausdorff
dimension
that
avoids the zeros of
for every
.
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
, the
number of input variables.
