Improved Upper Bounds on Key Invariants of Erd\H{o}s-R\'enyi Numerical Semigroups

TL;DR

This paper improves upper bounds on key invariants of Erdős-Rényi numerical semigroups using probabilistic methods, bringing them closer to known lower bounds, and introduces new results on sumsets in cyclic groups.

Contribution

It provides tighter upper bounds on Frobenius number and embedding dimension for random numerical semigroups, advancing understanding of their typical properties.

Findings

Upper bounds on Frobenius number improved to within polylogarithmic factor of lower bounds.

Upper bounds on embedding dimension similarly improved.

Proved that for random subsets of cyclic groups, the k-fold sumset covers the entire group with high probability.

Abstract

De Loera, O'Neill and Wilburne introduced a general model for random numerical semigroups in which each positive integer is chosen independently with some probability p to be a generator, and proved upper and lower bounds on the expected Frobenius number and expected embedding dimensions. We use a range of probabilistic methods to improve the upper bounds to within a polylogarithmic factor of the lower bounds in each case. As one of the tools to do this, we prove that for any prime q, if A is a random subset of the cyclic group Z_q whose size is of order log(q) and k is also of order log(q), then with high probability the k-fold sumset kA is all of Z_q.