Numbers Whose Powers Are Arbitrarily Close to Integers
In this paper, it is proved that, for any sequence of positive numbers <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>ξ</mi><mi>n</mi></msub></semantics></math&g...
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| Format: | Article |
| Language: | English |
| Published: |
MDPI AG
2025-05-01
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| Series: | Axioms |
| Subjects: | |
| Online Access: | https://www.mdpi.com/2075-1680/14/6/420 |
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| Summary: | In this paper, it is proved that, for any sequence of positive numbers <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>ξ</mi><mi>n</mi></msub></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi><mo>=</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mo>…</mo></mrow></semantics></math></inline-formula>, which does not converge to zero faster than the exponential function, and any sequence of positive numbers <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>δ</mi><mi>n</mi></msub></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi><mo>=</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mn>3</mn><mo>,</mo><mo>…</mo></mrow></semantics></math></inline-formula>, there is an uncountable set of positive numbers <i>S</i> such that, for each <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>α</mi><mo>></mo><mn>1</mn></mrow></semantics></math></inline-formula> in <i>S</i>, there are infinitely many <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi><mo>∈</mo><mi mathvariant="double-struck">N</mi></mrow></semantics></math></inline-formula> for which the fractional parts <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>{</mo><msub><mi>ξ</mi><mi>n</mi></msub><msup><mi>α</mi><mi>n</mi></msup><mo>}</mo></mrow></semantics></math></inline-formula> are smaller than <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>δ</mi><mi>n</mi></msub></semantics></math></inline-formula>, regardless of how fast the sequence <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>δ</mi><mi>n</mi></msub></semantics></math></inline-formula> tends to zero. In particular, for any sequence bounded away from zero, namely, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>ξ</mi><mi>n</mi></msub><mo>≥</mo><mi>ξ</mi><mo>></mo><mn>0</mn></mrow></semantics></math></inline-formula> for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi><mo>≥</mo><mn>1</mn></mrow></semantics></math></inline-formula>, it is shown that infinitely many integers <i>n</i> for which the inequality <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mrow><mo>{</mo><msub><mi>ξ</mi><mi>n</mi></msub><msup><mi>α</mi><mi>n</mi></msup><mo>}</mo></mrow><mo><</mo><msub><mi>δ</mi><mi>n</mi></msub></mrow></semantics></math></inline-formula> is true can be extracted from an arbitrary subsequence <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">N</mi></semantics></math></inline-formula> of positive integers. |
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| ISSN: | 2075-1680 |