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

Qian Xiao, Chao Ma and Shuailing Wang
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Qian Xiao: School of Mathematics, South China University of Technology, Guangzhou 510640, China
Chao Ma: Faculty of Information Technology, Macau University of Science and Technology, Macau 999078, China
Shuailing Wang: School of Mathematics, South China University of Technology, Guangzhou 510640, China

Mathematics, 2022, vol. 10, issue 1, 1-11

Abstract: In this paper, we consider continued β -fractions with golden ratio base β . We show that if the continued β -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 [ β ] 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 Lévy constants of the points with eventually periodic continued β -fraction expansion is dense in [ c , + ∞ ), where c = 1 2 log β + 2 − 5 β + 1 2 .

Keywords: ? -integer; continued ? -fractions; eventually periodic; Lévy constant (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2022
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