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<p>Let <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(\alpha \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>α</mi> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(\beta \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>β</mi> </math></EquationSource> </InlineEquation> be real algebraic numbers with <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\(\alpha \in [0,1]\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>α</mi> <mo>∈</mo> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">]</mo> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq6"> <EquationSource Format="TEX">\(1&lt; \beta \le 2\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>1</mn> <mo>&lt;</mo> <mi>β</mi> <mo>≤</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>. In this paper, we prove that the <InlineEquation ID="IEq7"> <EquationSource Format="TEX">\(\beta \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>β</mi> </math></EquationSource> </InlineEquation>-expansion of <InlineEquation ID="IEq8"> <EquationSource Format="TEX">\(\alpha \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>α</mi> </math></EquationSource> </InlineEquation> contains infinitely many occurrences of 7/3-powers. Our proof relies on Mahler’s method developed by Loxton and van der Poorten and the matryoshka structure theorem of infinite binary 7/3-power-free words obtained by Karhumäki and Shallit.</p>

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On patterns occurring in binary algebraic \(\beta \)-expansions of algebraic numbers

  • 期刊论文

  • 发表日期：

    2026年2月18日

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  • Eiji Miyanohara

摘要

Let \(\alpha \) α and \(\beta \) β be real algebraic numbers with \(\alpha \in [0,1]\) α ∈ [ 0 , 1 ] and \(1< \beta \le 2\) 1 < β ≤ 2 . In this paper, we prove that the \(\beta \) β -expansion of \(\alpha \) α contains infinitely many occurrences of 7/3-powers. Our proof relies on Mahler’s method developed by Loxton and van der Poorten and the matryoshka structure theorem of infinite binary 7/3-power-free words obtained by Karhumäki and Shallit.