Ulam Sequences and Ulam Sets

2017/05/04 by Noah Kravitz, Stefan Steinerberger, Kravitz, Noah +1 · 1 citation
Computer Science · Mathematics · #Advanced Combinatorial Mathematics #Combinatorics (math.CO) #Discrete Mathematics (cs.DM) #Dynamical Systems (math.DS) #FOS: Computer and information sciences #FOS: Mathematics #Mathematical Dynamics and Fractals #semigroups and automata theory

paper · pdf · doi:10.48550/arxiv.1705.01883

openalex publication_date 2017/05/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The Ulam sequence is given by a₁ =1, a₂ = 2, and then, for n ≥ 3, the element a_n is defined as the smallest integer that can be written as the sum of two distinct earlier elements in a unique way. This gives the sequence 1, 2, 3, 4, 6, 8, 11, 13, 16, …, which has a mysterious quasi-periodic behavior that is not understood. Ulam's definition naturally extends to higher dimensions: for a set of initial vectors \v₁, …, v_k\ ⊂ ℝⁿ, we define a sequence by repeatedly adding the smallest elements that can be uniquely written as the sum of two distinct vectors already in the set. The resulting sets have very rich structure that turns out to be universal for many commuting binary operations. We give examples of different types of behavior, prove several universality results, and describe new unexplained phenomena.

Citations

• The Unreasonable Rigidity of Ulam Sets

Cited by

• Arithmetical Complexity and Absoluteness of Rigidity Phenomena for Ulam Sequences

Related

• Ulam Sets in New Settings
• Small gaps in the Ulam sequence
• The Ulam Sequence of the Integer Polynomial Ring
• A Hidden Signal in the Ulam sequence
• The Unreasonable Rigidity of Ulam Sets