On the Eventually Periodic Continued β-Fractions and Their Levy Constants

In this paper, we consider continued beta-fractions with golden ratio base beta. We show that if the continued beta-fraction expansion of a non-negative real number is eventually periodic, then it is the root of a quadratic irreducible polynomial with the coefficients in Z[beta] and we conjecture the converse is false, which is different from Lagrange's theorem for the regular continued fractions. We prove that the set of Levy constants of the points with eventually periodic continued beta-fraction expansion is dense in [c, +& INFIN;), where c=1/2log beta+2-root 5 beta+1/2.

Automated summary

In this paper, we study continued beta-fractions with golden ratio base beta and prove that if the continued beta-fraction expansion of a non-negative real number is periodic, then it is the root of a quadratic irreducible polynomial with coefficients in Z[beta].