Minimal zero entropy subshifts can be unrestricted along any sparse set

Abstract We present a streamlined proof of a result essentially presented by the author in Some counterexamples in topological dynamics. Ergod. Th. & Dynam. Sys. 28 (4) (2008), 1291–1322, namely that for every set S = \s₁, s₂, \ N of zero Banach density and finite set A, there exists a minimal zero-entropy subshift (X, ) so that for every sequence u A^ {Z}, there is xᵤ X with xᵤ (sₙ) = u (n) for all n 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 reported by Eisner A polynomial version of Sarnak’s conjecture. C. R. Math. Acad. Sci. Paris 353 (7) (2015), 569–572 which are significantly more general than some recently provided by Kanigowski, Lemańczyk and Radziwiłł Prime number theorem for analytic skew products. Ann. of Math. (2) 199 (2024), 591–705 and by Lian and Shi A counter-example for polynomial version of Sarnak’s conjecture. Adv. Math. 384 (2021), Paper no. 107765 and shows that no similar result can hold under only the assumptions of minimality and zero entropy.