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...
Saved in:
| Main Author: | |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
MDPI AG
2025-05-01
|
| Series: | Axioms |
| Subjects: | |
| Online Access: | https://www.mdpi.com/2075-1680/14/6/420 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1849467405672120320 |
|---|---|
| author | Artūras Dubickas |
| author_facet | Artūras Dubickas |
| author_sort | Artūras Dubickas |
| collection | DOAJ |
| description | 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. |
| format | Article |
| id | doaj-art-e1efb01ffb4d48a8a49c4bc091edcb39 |
| institution | Kabale University |
| issn | 2075-1680 |
| language | English |
| publishDate | 2025-05-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Axioms |
| spelling | doaj-art-e1efb01ffb4d48a8a49c4bc091edcb392025-08-20T03:26:15ZengMDPI AGAxioms2075-16802025-05-0114642010.3390/axioms14060420Numbers Whose Powers Are Arbitrarily Close to IntegersArtūras Dubickas0Institute of Mathematics, Faculty of Mathematics and Informatics, Vilnius University, Naugarduko 24, LT-03225 Vilnius, LithuaniaIn 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.https://www.mdpi.com/2075-1680/14/6/420fractional partspowers of transcendental numbersdistribution modulo 1 |
| spellingShingle | Artūras Dubickas Numbers Whose Powers Are Arbitrarily Close to Integers Axioms fractional parts powers of transcendental numbers distribution modulo 1 |
| title | Numbers Whose Powers Are Arbitrarily Close to Integers |
| title_full | Numbers Whose Powers Are Arbitrarily Close to Integers |
| title_fullStr | Numbers Whose Powers Are Arbitrarily Close to Integers |
| title_full_unstemmed | Numbers Whose Powers Are Arbitrarily Close to Integers |
| title_short | Numbers Whose Powers Are Arbitrarily Close to Integers |
| title_sort | numbers whose powers are arbitrarily close to integers |
| topic | fractional parts powers of transcendental numbers distribution modulo 1 |
| url | https://www.mdpi.com/2075-1680/14/6/420 |
| work_keys_str_mv | AT arturasdubickas numberswhosepowersarearbitrarilyclosetointegers |