An exponential bound for generalized numerical semigroups of fixed dimension

Abstract

Let n(g,d) be the number of cofinite additive submonoids of ℕ₀ᵈ with complement of cardinality g. For each fixed d, we prove n(g,d) = 2^O_d(g), answering Question 49 of Li. The main estimate bounds the entire coordinatewise downclosure of the complement by 4^(d+1)g. We apply classical maximal-average estimates to the local density supplied by reflection, and give an elementary finite proof using Abel summation. A closed-word encoding of downsets gives an explicit exponential bound. The core argument is formalized in Lean. A separate, not yet formalized section gives a smaller upper growth constant using elementary downset estimates weaker than known results in multidimensional partition theory.

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