For an irrational number <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\alpha $</EquationSource> <EquationSource Format="MATHML"><math> <mi>α</mi> </math></EquationSource> </InlineEquation>, it is well-known that the fractional part of <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$m\alpha $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>m</mi> <mi>α</mi> </mrow> </math></EquationSource> </InlineEquation> is dense in the interval (0, 1). This in particular implies that an integral multiple of <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\alpha $</EquationSource> <EquationSource Format="MATHML"><math> <mi>α</mi> </math></EquationSource> </InlineEquation> captures a given digit in base b at least once. In 1973, Mahler proved that some integral multiple of a given irrational number <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\alpha $</EquationSource> <EquationSource Format="MATHML"><math> <mi>α</mi> </math></EquationSource> </InlineEquation> captures a given block B of length n in base b infinitely often. However, the integer in Mahler’s result is existential and lies in an interval <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$[1, 2b^{n+1}]$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">[</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <msup> <mi>b</mi> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">]</mo> </mrow> </math></EquationSource> </InlineEquation>. In this short note, we explicitly find an interval I such that for all integers <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$X\in I$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>X</mi> <mo>∈</mo> <mi>I</mi> </mrow> </math></EquationSource> </InlineEquation>, the given block B of length n in base b occurs in the fractional part of <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$X\alpha $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>X</mi> <mi>α</mi> </mrow> </math></EquationSource> </InlineEquation> with some frequency, under the assumptions of appearance of block of zeroes in <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$\alpha $</EquationSource> <EquationSource Format="MATHML"><math> <mi>α</mi> </math></EquationSource> </InlineEquation>. Without these assumptions, finding the explicit values of integer X seems a difficult problem.

A note on Mahler’s theorem – II

期刊论文
发表日期：

2025年11月27日

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R Thangadurai,
Aparna Tripathi

摘要

For an irrational number $\alpha $ $α$ , it is well-known that the fractional part of $m\alpha $ $m α$ is dense in the interval (0, 1). This in particular implies that an integral multiple of $\alpha $ $α$ captures a given digit in base b at least once. In 1973, Mahler proved that some integral multiple of a given irrational number $\alpha $ $α$ captures a given block B of length n in base b infinitely often. However, the integer in Mahler’s result is existential and lies in an interval $[1, 2b^{n+1}]$ $[1, 2 b^{n + 1}]$ . In this short note, we explicitly find an interval I such that for all integers $X\in I$ $X \in I$ , the given block B of length n in base b occurs in the fractional part of $X\alpha $ $X α$ with some frequency, under the assumptions of appearance of block of zeroes in $\alpha $ $α$ . Without these assumptions, finding the explicit values of integer X seems a difficult problem.