2017/10/31 by Joshua Hinman, Borys Kuca, Hinman, Joshua +5
Computer Science · Mathematics · #03C98 (Primary) #05A16 (Secondary) #11B83 #Advanced Topology and Set Theory #Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT) #Rings, Modules, and Algebras #semigroups and automata theory
paper · pdf · doi:10.48550/arxiv.1711.00145
openalex publication_date 2017/10/31 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We give a number of results about families of Ulam sets. Generalizing behavior of Ulam sets U(1,n), we prove using an novel model theoretic approach that there is a rigidity phenomenon for Ulam sets U(a,b) as b increases. Based on this, we suggest a natural conjecture, and investigate its potential applications, including a method of proving certain families of Ulam sequences are regular, for which we also provide partial, unconditional, results. Along this same vein, we give an upper bound bound on the density of Ulam sequences U(1,n). Finally, we give classification results for higher dimensional Ulam sets.