Can one construct an infinite recursively enumerable set \(S\subset \mathbb{N}\), for which one can decide: given any linear recurrence sequence \((u_n)\), whether \(\exists n\in S\), s.t. \(u_n<0\) ? < p>

In other words: is there a set of indices for which the positivity problem is decidable?

The analogous question where \(u_n<0\) is replaced by \(u_n="0\)" has a positive answer for simple sequences; luca, ouaknine and worrell have constructed such set explicitly (called the universal skolem set). < p>