Counting numerical sets with no small atoms

A numerical set S with Frobenius number g is a set of integers with min ( S ) = 0 and max ( Z − S ) = g , and its atom monoid is A ( S ) = { n ∈ Z | n + s ∈ S for all s ∈ S } . Let γ g be the ratio of the number of numerical sets S having A ( S ) = { 0 } ∪ ( g , ∞ ) divided by the total number of numerical sets with Frobenius number g. We show that the sequence { γ g } is decreasing and converges to a number γ ∞ ≈ . 4844 (with accuracy to within .0050). We also examine the singularities of the generating function for { γ g } . Parallel results are obtained for the ratio γ g σ of the number of symmetric numerical sets S with A ( S ) = { 0 } ∪ ( g , ∞ ) by the number of symmetric numerical sets with Frobenius number g. These results yield information regarding the asymptotic behavior of the number of finite additive 2-bases.