  We present a streamlined proof of a result essentially present in previous
work of the author, namely that for every set $S = \{s_1, s_2, \ldots\} \subset
\mathbb{N}$ of zero Banach density and finite set $A$, there exists a minimal
zero-entropy subshift $(X, \sigma)$ so that for every sequence $u \in
A^\mathbb{Z}$, there is $x_u \in X$ with $x_u(s_n) = u(n)$ for all $n \in
\mathbb{N}$. Informally, minimal deterministic sequences can achieve completely
arbitrary behavior upon restriction to a set of zero Banach density.
  As a corollary, this provides counterexamples to the Polynomial Sarnak
Conjecture which are significantly more general than some recently provided in
word of Kanigowski, Lema\'{n}czyk, and Radziwi\l\l and of Lian and Shi, and
shows that no similar result can hold under only the assumptions of minimality
and zero entropy.
