On simultaneous approximation to (α,α2) with α3+kα−1=0

AbstractWe show that the modified Jacobi–Perron algorithm gives the best simultaneous approximation to (α,α2) with α3+kα−1=0. We claim the following facts: (1)the limit set of {(qn(qnα−pn),qn(qnα2−rn)|n=1,2,…} become an ellipse, where (pn,qn,rn) is the nth convergent (pn/qn,rn/qn) of (α,α2) by the modified Jacobi–Perron algorithm,(2)the limit set of {(q(qα−p),q(qα2−r)|q∈Z,q>0} belongs to outside of the ellipse in (1)

Dual substitutions over mathbbR>0−mathbb{R}_{>0}-powered symbols (Natural extension of arithmetic algorithms and S-adic system)

"Natural extension of arithmetic algorithms and S-adic system". July 20～24, 2015. edited by Shigeki Akiyama. The papers presented in this volume of RIMS Kôkyûroku Bessatsu are in final form and refereed.We will consider dual substitutions of substitutions over R>0-powered symbols which are introduced in [12]. We give a class of weighted substitutions including the dual substitutions. We also introduce weighted tips which generalize the unit tip in a stepped surface (discrete plane)

Substitutions over mathbbCmathbb{C}-powered symbols, and Rauzy fractals for imaginary directions (Natural extension of arithmetic algorithms and S-adic system)

"Natural extension of arithmetic algorithms and S-adic system". July 20～24, 2015. edited by Shigeki Akiyama. The papers presented in this volume of RIMS Kôkyûroku Bessatsu are in final form and refereed.本文ファイルではp.1-39と表記We introduce substitutions over complex powered symbols az(zEIC), and extend the definition of the Rauzy fractal and give some new examples ofRauzy fractals. We also give some problems and results related to simultaneous Diophantine approximations and multidimensional complex continued fractions