Linear Recurrence Sequences (LRS) are a fundamental mathematical primitive for a plethora of applications such as model checking, probabilistic systems, computational biology, and economics. Positivity (are all terms of the given LRS at least 0?) and Ultimate Positivity (are all but finitely many terms of the given LRS at least 0?) 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. In this paper, we consider Robust Positivity and Ultimate Positivity problems where the neighbourhood of the initialisation, specified 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 can't handle would entail significant number-theoretic breakthroughs.

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