Numerical Fractal Analysis of Exceptional Sets in the Lehner Expansion

Abstract

This paper examines how the average value of the sequence bn in the Lehner expansion of a real number x influences its box dimension. Our primary objective is to analyze how variations in the average of bn impact the box dimension, which serves as a measure of the complexity of the sequence. Using the box-counting method, we numerically estimate the box dimension and explore its relationship with the fractal nature of Lehner expansions.

More by Shirali Kadyrov

• Bounded continued fractions and quadratic orders in $\Q(\sqrt{127})$arXiv · 2026
• Code and data for "Reinforcement learning-based PID for nonlinear temperature control with delayed feedback and measurement noise"Zenodo · 2026
• Life expectancy in Kazakhstan: an integrated analysis and forecastZenodo · 2026
• Data-driven homotopic reinforcement learning for adaptive tracking control of completely unknown Takagi–Sugeno fuzzy Markovian jump systemsCommunications in Nonlinear Science and Numerical Simulation · 2026