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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
: ( ℝ n ) v
→ 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