Sparse regular subsets of the reals

2023/11/18 by Jason P. Bell, Bell, Jason, Alexi Block Gorman +1
Computer Science · Mathematics · #03C64 #03D05 #28A80 #Advanced Topology and Set Theory #Computability, Logic, AI Algorithms #FOS: Computer and information sciences #FOS: Mathematics #Formal Languages and Automata Theory (cs.FL) #Logic (math.LO) #Mathematical Dynamics and Fractals

paper · pdf · doi:10.48550/arxiv.2311.11162

openalex publication_date 2023/11/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01

Abstract

This paper concerns the expansion of the real ordered additive group by a predicate for a subset of [0,1] whose base-r representations are recognized by a Büchi automaton. In the case that this predicate is closed, a dichotomy is established for when this expansion is interdefinable with the structure (ℝ,<,+,0,r^-ℕ) for some r ∈ ℕ_>1. In the case that the closure of the predicate has Hausdorff dimension less than 1, the dichotomy further characterizes these expansions of (ℝ,<,+,0,1) by when they have NIP and NTP₂, which is precisely when the closure of the predicate has Hausdorff dimension 0.

Related

• Structure theorem for i-minimal expansions of the real additive ordered group
• A tetrachotomy for expansions of the real ordered additive group
• Büchi automata recognizing sets of reals definable in first-order logic with addition and order
• Fractal dimensions of k-automatic sets
• Contributions to the theory of F-automatic sets