  We study the decidability of the Skolem Problem, the Positivity Problem, and
the Ultimate Positivity Problem for linear recurrences with real number initial
values and real number coefficients in the bit-model of real computation. We
show that for each problem there exists a correct partial algorithm which halts
for all problem instances for which the answer is locally constant, thus
establishing that all three problems are as close to decidable as one can
expect them to be in this setting. We further show that the algorithms for the
Positivity Problem and the Ultimate Positivity Problem halt on almost every
instance with respect to the usual Lebesgue measure on Euclidean space. In
comparison, the analogous problems for exact rational or real algebraic
coefficients are known to be decidable only for linear recurrences of fairly
low order.
