  The Skolem problem is a long-standing open problem in linear dynamical
systems: can a linear recurrence sequence (LRS) ever reach 0 from a given
initial configuration? Similarly, the positivity problem asks whether the LRS
stays positive from an initial configuration. Deciding Skolem (or positivity)
has been open for half a century: the best known decidability results are for
LRS with special properties (e.g., low order recurrences). But these problems
are easier for "uninitialized" variants, where the initial configuration is not
fixed but can vary arbitrarily: checking if there is an initial configuration
from which the LRS stays positive can be decided in polynomial time (Tiwari in
2004, Braverman in 2006). In this paper, we consider problems that lie between
the initialized and uninitialized variants. More precisely, we ask if 0 (resp.
negative numbers) can be avoided from every initial configuration in a
neighborhood of a given initial configuration. This can be considered as a
robust variant of the Skolem (resp. positivity) problem. We show that these
problems lie at the frontier of decidability: if the neighbourhood is given as
part of the input, then robust Skolem and robust positivity are Diophantine
hard, i.e., solving either would entail major breakthroughs in Diophantine
approximations, as happens for (non-robust) positivity. However, if one asks
whether such a neighbourhood exists, then the problems turn out to be decidable
with PSPACE complexity. Our techniques also allow us to tackle robustness for
ultimate positivity, which asks whether there is a bound on the number of steps
after which the LRS remains positive. There are two variants depending on
whether we ask for a "uniform" bound on this number of steps. For the
non-uniform variant, when the neighbourhood is open, the problem turns out to
be tractable, even when the neighbourhood is given as input.
