On the weakness of the vague specification property

We show that the vague specification property is strictly weaker than most of the specification-like properties, by establishing its equivalence with the asymptotic average shadowing property. In particular, we see that the weak specification property implies the vague specification property, but the converse does not hold, answering the question posed by Downarowicz and Weiss in [Ergod. Th. \& Dynam. Sys. 44(9) (2024), 2565--2580]. Additionally, we prove that, for surjective systems, the asymptotic average shadowing property is equivalent to the average shadowing property if the phase space is complete with respect to the dynamical Besicovitch pseudometric. We use the combination of both results to prove that the proximal and minimal shift spaces from [Ergod. Th. \& Dynam. Sys., 45(2) (2025), 396--426] possess the vague specification property (asymptotic average shadowing property). Our findings also allow us to address a couple of questions from [Fund. Math., 224(3) (2014), 241--278] about the asymptotic average shadowing property.