A Dombi Counterexample with Positive Lower Density

2023/02/04 by Jeffrey Shallit, Shallit, Jeffrey
Computer Science · Mathematics · #semigroups and automata theory #Advanced Algebra and Logic #Limits and Structures in Graph Theory

paper · pdf · doi:10.48550/arxiv.2302.02138

Abstract

Let r(k,A,n) denote the number of representations of n as a sum of k elements of a set A ⊆ ℕ. In 2002, Dombi conjectured that if A is co-infinite, then the sequence (r(k,A,n))_{n ≥ 0} cannot be strictly increasing. Using tools from automata theory and logic, we give an explicit counterexample where ℕ ∖ A has positive lower density.

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