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

Tristram Bogart; Santiago Morales

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arXiv · 2411.13767

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

Tristram Bogart·Santiago Morales

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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.

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Tristram Bogart, Santiago Morales. 2024-11-21. Improved Upper Bounds on Key Invariants of Erd\H{o}s-R\'enyi Numerical Semigroups. https://arxiv.org/abs/2411.13767

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