Exclude clumped Ulam elements followed by a large gap

Determine how to exclude, for the Ulam sequence, the scenario in which sequence elements occur in clumped groups followed by a very large gap, thereby resolving the principal obstruction to deriving stronger growth conclusions from the small-gap strategy.

Background

The proof of Theorem 2 shows that rapid growth would create many candidate sums of the form a_n + a_k near a_n, while comparatively few sums of two earlier terms could block them. The authors identify a possible obstruction: Ulam numbers might occur in tightly grouped clusters followed by a large jump, allowing all candidate sums to be blocked. They explicitly state that they do not know how to rule out this configuration.