The Unreasonable Rigidity of Ulam Sets

TL;DR

This paper explores the structural rigidity of Ulam sets, introduces a model theoretic approach to analyze their behavior, and provides new bounds, conjectures, and classifications for these mathematical objects.

Contribution

It introduces a novel model theoretic approach to study Ulam sets and establishes a rigidity phenomenon for Ulam sets U(a,b) as b increases, along with new bounds and classifications.

Findings

Rigidity phenomenon for Ulam sets U(a,b) as b increases

Upper bound on the density of Ulam sequences U(1,n)

Classification results for higher dimensional Ulam sets

Abstract

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.