Probability that an early m-fold sumset contains a small prime

Ascertain a quantitative estimate for the probability that, in the unconstrained Erdős–Rényi-type random numerical semigroup model (p), the m-fold sumset G of the first k selected generators g1, …, gk (with k a fixed constant and each gi selected in {1, …, 2k/p}) contains at least one prime when m is slightly larger than log log(1/p).

Background

To reduce logarithmic factors in the upper bounds, the authors suggest leveraging early selected small generators by forming a modest m-fold sumset and hoping it contains a prime that could replace the large prime q used in their main argument.

They explicitly state uncertainty about estimating the probability that such a sumset contains a small prime; resolving this would enable sharper bounds and potentially eliminate one log factor in their results.