2023/05/29 by Fumichika Takamizo, Takamizo, Fumichika · 2 citations
Mathematics · Computer Science · #Mathematical Dynamics and Fractals #Computability, Logic, AI Algorithms #Advanced Topology and Set Theory
paper · pdf · doi:10.48550/arxiv.2306.01770
Let β>1. For x ∈ [0,∞), we have so-called the β-expansion of x in base β as follows: x= ∑j ≤ k xjβj = xkβk+ ⋯ + x1β+x0+x-1β-1 + x-2β-2 + ⋯ where k ∈ ℤ, βk ≤ x < βk+1, xj ∈ ℤ ∩ [0,β) for all j ≤ k and ∑j ≤ nxjβj<βn+1 for all n ≤ k. In this paper, we give a sufficient condition (for β) such that each element of ℕ has the finite beta-expansion in base β. Moreover we also find a β with this finiteness property which does not have positive finiteness property.