2014/11/18 by Simon Baker, Baker, Simon · 1 citation
Computer Science · Mathematics · #11K06 #11K31 #Advanced Topology and Set Theory #Computability, Logic, AI Algorithms #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals #Number Theory (math.NT)
paper · doi:10.48550/arxiv.1411.4817
openalex publication_date 2014/11/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Given a strictly increasing sequence of positive real numbers tending to infinity (qn)n=1∞, and an arbitrary sequence of real numbers (rn)n=1∞. We study the set of α∈(1,∞) for which limn→∞‖α^qn-rn‖= 0. In \citeDub Dubickas showed that whenever limn→∞(qn+1-qn)=∞, there always exists a transcendental α for which limn→∞‖α^qn-rn‖= 0. Adapting the approach of Bugeaud and Moshchevitin \citeBugMos, we improve upon this result and show that whenever limn→∞(qn+1-qn)=∞, the set of α∈(1,∞) satisfying limn→∞‖α^qn-rn‖= 0 is a dense set of Hausdorff dimension 1.