{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T01:19:55Z","timestamp":1760059195799,"version":"build-2065373602"},"reference-count":32,"publisher":"MDPI AG","issue":"6","license":[{"start":{"date-parts":[[2025,5,29]],"date-time":"2025-05-29T00:00:00Z","timestamp":1748476800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>In this paper, it is proved that, for any sequence of positive numbers \u03ben, n=1,2,\u2026, which does not converge to zero faster than the exponential function, and any sequence of positive numbers \u03b4n, n=1,2,3,\u2026, there is an uncountable set of positive numbers S such that, for each \u03b1&gt;1 in S, there are infinitely many n\u2208N for which the fractional parts {\u03ben\u03b1n} are smaller than \u03b4n, regardless of how fast the sequence \u03b4n tends to zero. In particular, for any sequence bounded away from zero, namely, \u03ben\u2265\u03be&gt;0 for n\u22651, it is shown that infinitely many integers n for which the inequality {\u03ben\u03b1n}&lt;\u03b4n is true can be extracted from an arbitrary subsequence N of positive integers.<\/jats:p>","DOI":"10.3390\/axioms14060420","type":"journal-article","created":{"date-parts":[[2025,5,29]],"date-time":"2025-05-29T07:46:26Z","timestamp":1748504786000},"page":"420","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Numbers Whose Powers Are Arbitrarily Close to Integers"],"prefix":"10.3390","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-3625-9466","authenticated-orcid":false,"given":"Art\u016bras","family":"Dubickas","sequence":"first","affiliation":[{"name":"Institute of Mathematics, Faculty of Mathematics and Informatics, Vilnius University, Naugarduko 24, LT-03225 Vilnius, Lithuania"}]}],"member":"1968","published-online":{"date-parts":[[2025,5,29]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"313","DOI":"10.1007\/BF01475864","article-title":"\u00dcber die Gleichverteilung von Zahlen modulo Eins","volume":"77","author":"Weyl","year":"1916","journal-title":"Math. 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