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<p>Extending a well-known classical result on admissible words given by Rényi in 1957, for any <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\beta &gt;1\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>β</mi> <mo>&gt;</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>, we prove that the number of full words with length <i>n</i> in <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\beta \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>β</mi> </math></EquationSource> </InlineEquation>-expansions is comparable to <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(\beta ^n\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>β</mi> <mi>n</mi> </msup> </math></EquationSource> </InlineEquation>. When <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(\beta &gt;1\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>β</mi> <mo>&gt;</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> is not an integer, the conclusion is also true for the non-full words. The proofs are based on a technical decomposition for the set of full words with the same length.</p>

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Numbers of full and non-full words in beta-expansions

  • 期刊论文

  • 发表日期：

    2025年10月8日

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  • Yao-Qiang Li

摘要

Extending a well-known classical result on admissible words given by Rényi in 1957, for any \(\beta >1\) β > 1 , we prove that the number of full words with length n in \(\beta \) β -expansions is comparable to \(\beta ^n\) β n . When \(\beta >1\) β > 1 is not an integer, the conclusion is also true for the non-full words. The proofs are based on a technical decomposition for the set of full words with the same length.