Biased partitions of $\mathbb{Z}^n$

Given a function $f$ on the vertex set of some graph $G$, a scenery, let a simple random walk run over the graph and produce a sequence of values. Is it possible to, with high probability, reconstruct the scenery $f$ from this random sequence? To show this is impossible for some graphs, Gross and Grupel, call a function $f:V\to\{0,1\}$ on the vertex set of a graph $G=(V,E)$ $p$-biased if for each vertex $v$ the fraction of neighbours on which $f$ is 1 is exactly $p$. Clearly, two $p$-biased functions are indistinguishable based on their sceneries. Gross and Grupel construct $p$-biased functions on the hypercube $\{0,1\}^n$ and ask for what $p\in[0,1]$ there exist $p$-biased functions on $\mathbb{Z}^n$ and additionally how many there are. We fully answer this question by giving a complete characterization of these values of $p$. We show that $p$-biased functions exist for all $p=c/2n$ with $c\in\{0,\dots,2n\}$ and, in fact, there are uncountably many of them for every $c\in\{1,\dots,2n-1\}$. To this end, we construct uncountably many partitions of $\mathbb{Z}^n$ into $2n$ parts such that every element of $\mathbb{Z}^n$ has exactly one neighbour in each part. This additionally shows that not all sceneries on $\mathbb{Z}^n$ can be reconstructed from a sequence of values on attained on a simple random walk.

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