Locally biased partitions of Zn

Abstract 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, in Gross and Grupel (2018), call a function f : V → { 0 , 1 } on the vertex set of a graph G = ( V , E ) locally p -biased if for each vertex v the fraction of neighbours on which f is 1 is exactly p . Clearly, two locally p -biased functions are indistinguishable based on their sceneries. Gross and Grupel construct locally p -biased functions on the hypercube { 0 , 1 } n and ask for what p ∈ [ 0 , 1 ] there exist locally p -biased functions on 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 locally p -biased functions exist for all p = c ∕ 2 n with c ∈ { 0 , … , 2 n } and that, in fact, there are uncountably many of them for every c ∈ { 1 , … , 2 n − 1 } . To this end, we construct uncountably many partitions of Z n into 2 n parts such that every element of Z n has exactly one neighbour in each part. This additionally shows that not all sceneries on Z n can be reconstructed from a sequence of values attained on a simple random walk.