  Linear Recurrence Sequences (LRS) are a fundamental mathematical primitive
for a plethora of applications such as the verification of probabilistic
systems, model checking, computational biology, and economics. Positivity (are
all terms of the given LRS non-negative?) and Ultimate Positivity (are all but
finitely many terms of the given LRS non-negative?) are important open
number-theoretic decision problems. Recently, the robust versions of these
problems, that ask whether the LRS is (Ultimately) Positive despite small
perturbations to its initialisation, have gained attention as a means to model
the imprecision that arises in practical settings. However, the state of the
art is ill-equipped to reason about imprecision when its extent is explicitly
specified. In this paper, we consider Robust Positivity and Ultimate Positivity
problems where the neighbourhood of the initialisation, expressed in a natural
and general format, is also part of the input. We contribute by proving sharp
decidability results: decision procedures at orders our techniques are unable
to handle for general LRS would entail significant number-theoretic
breakthroughs.
