| Summary: | In this paper, we consider continued <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>β</mi></semantics></math></inline-formula>-fractions with golden ratio base <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>β</mi></semantics></math></inline-formula>. We show that if the continued <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>β</mi></semantics></math></inline-formula>-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 <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="double-struck">Z</mi><mo>[</mo><mi>β</mi><mo>]</mo></mrow></semantics></math></inline-formula> 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 <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>β</mi></semantics></math></inline-formula>-fraction expansion is dense in [<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="bold">c</mi></semantics></math></inline-formula>, +<i>∞</i>), where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="bold">c</mi><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo form="prefix">log</mo><mfrac><mrow><mi>β</mi><mo>+</mo><mn>2</mn><mo>−</mo><msqrt><mrow><mn>5</mn><mi>β</mi><mo>+</mo><mn>1</mn></mrow></msqrt></mrow><mn>2</mn></mfrac></mrow></semantics></math></inline-formula>.
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