Asymptotic second-order constant for random subset-sum coverage

Determine the constant c_* defined by c_* = limsup_{N→∞} (f(N) − log_2 N) / log log N, where f(N) is the minimal integer k such that, for every finite abelian group G of order N and a uniformly random k-element subset A ⊆ G, the subset-sum set {∑_{x∈S} x : S ⊆ A} equals G with probability at least 1/2.

Background

Let f(N) denote the least k such that a uniformly random k-element subset A of a finite abelian group G of order N has its subset-sum set equal to G with probability at least 1/2. Erdős and Rényi proved the universal upper bound f(N) ≤ log_2 N + (1/ log 2) log log N + O(1).

This paper proves a quantitative lower bound for prime orders p, showing f(p) ≥ log_2 p + (1/(2 log 2) + o(1)) log log p, thereby ruling out the bound f(N) ≤ log_2 N + o(log log N) in general. Motivated by sharpening the second-order term, the authors pose determining the limsup constant c_* as an explicit problem.

From the bounds in the literature and this paper, one obtains 1/(2 log 2) ≤ c_* ≤ 1/ log 2. The authors note heuristics and numerical evidence suggesting c_* = 1/ log 2, but they do not prove this, leaving the precise value of c_* open.