{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T20:07:23Z","timestamp":1776802043578,"version":"3.51.2"},"reference-count":13,"publisher":"American Mathematical Society (AMS)","issue":"302","license":[{"start":{"date-parts":[[2017,1,19]],"date-time":"2017-01-19T00:00:00Z","timestamp":1484784000000},"content-version":"am","delay-in-days":366,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"

\n Let\n \n \n \n \n I<\/mml:mi>\n <\/mml:mrow>\n \\mathbb {I}<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n be the field of rational numbers or an imaginary quadratic field and\n \n \n \n \n \n Z<\/mml:mi>\n <\/mml:mrow>\n \n I<\/mml:mi>\n <\/mml:mrow>\n <\/mml:msub>\n \\mathbb {Z}_\\mathbb {I}<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n its ring of integers. We study some general lemmas that produce lower bounds\n \n \\[\n \n \n \n \n |\n \n <\/mml:mo>\n \n B<\/mml:mi>\n 0<\/mml:mn>\n <\/mml:msub>\n +<\/mml:mo>\n \n B<\/mml:mi>\n 1<\/mml:mn>\n <\/mml:msub>\n \n \n \u03b8\n \n <\/mml:mi>\n 1<\/mml:mn>\n <\/mml:msub>\n +<\/mml:mo>\n \n \u22ef\n \n <\/mml:mo>\n +<\/mml:mo>\n \n B<\/mml:mi>\n r<\/mml:mi>\n <\/mml:msub>\n \n \n \u03b8\n \n <\/mml:mi>\n r<\/mml:mi>\n <\/mml:msub>\n \n |\n \n <\/mml:mo>\n \n \u2265\n \n <\/mml:mo>\n \n 1<\/mml:mn>\n \n max<\/mml:mo>\n {<\/mml:mo>\n \n |\n \n <\/mml:mo>\n \n B<\/mml:mi>\n 1<\/mml:mn>\n <\/mml:msub>\n \n |\n \n <\/mml:mo>\n ,<\/mml:mo>\n \n \u2026\n \n <\/mml:mo>\n ,<\/mml:mo>\n \n |\n \n <\/mml:mo>\n \n B<\/mml:mi>\n r<\/mml:mi>\n <\/mml:msub>\n \n |\n \n <\/mml:mo>\n \n }<\/mml:mo>\n \n \u03bc\n \n <\/mml:mi>\n <\/mml:msup>\n <\/mml:mrow>\n <\/mml:mfrac>\n <\/mml:mrow>\n \\lvert B_0+B_1\\theta _1+\\cdots +B_r\\theta _r\\rvert \\ge \\frac {1}{\\max \\{\\lvert B_1 \\rvert ,\\ldots ,\\lvert B_r \\rvert \\}^\\mu }<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n \\]\n <\/disp-formula>\n for all\n \n \n \n \n \n B<\/mml:mi>\n 0<\/mml:mn>\n <\/mml:msub>\n ,<\/mml:mo>\n \n \u2026\n \n <\/mml:mo>\n ,<\/mml:mo>\n \n B<\/mml:mi>\n r<\/mml:mi>\n <\/mml:msub>\n \n \u2208\n \n <\/mml:mo>\n \n \n Z<\/mml:mi>\n <\/mml:mrow>\n \n \n I<\/mml:mi>\n <\/mml:mrow>\n <\/mml:mrow>\n <\/mml:msub>\n <\/mml:mrow>\n B_0,\\ldots ,B_r \\in \\mathbb {Z}_{\\mathbb {I}}<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n ,\n \n \n \n \n max<\/mml:mo>\n {<\/mml:mo>\n \n |\n \n <\/mml:mo>\n \n B<\/mml:mi>\n 1<\/mml:mn>\n <\/mml:msub>\n \n |\n \n <\/mml:mo>\n ,<\/mml:mo>\n \n \u2026\n \n <\/mml:mo>\n ,<\/mml:mo>\n \n |\n \n <\/mml:mo>\n \n B<\/mml:mi>\n r<\/mml:mi>\n <\/mml:msub>\n \n |\n \n <\/mml:mo>\n }<\/mml:mo>\n \n \u2265\n \n <\/mml:mo>\n \n H<\/mml:mi>\n 0<\/mml:mn>\n <\/mml:msub>\n <\/mml:mrow>\n \\max \\{\\lvert B_1 \\rvert ,\\ldots ,\\lvert B_r \\rvert \\} \\ge H_0<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n , given suitable simultaneous approximating sequences of the numbers\n \n \n \n \n \n \n \u03b8\n \n <\/mml:mi>\n 1<\/mml:mn>\n <\/mml:msub>\n ,<\/mml:mo>\n \n \u2026\n \n <\/mml:mo>\n ,<\/mml:mo>\n \n \n \u03b8\n \n <\/mml:mi>\n r<\/mml:mi>\n <\/mml:msub>\n <\/mml:mrow>\n \\theta _1,\\ldots ,\\theta _r<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n . We manage to replace the lower bound with\n \n \n \n \n 1<\/mml:mn>\n \n \/<\/mml:mo>\n <\/mml:mrow>\n \n max<\/mml:mo>\n {<\/mml:mo>\n \n |\n \n <\/mml:mo>\n \n B<\/mml:mi>\n 1<\/mml:mn>\n <\/mml:msub>\n \n \n |\n \n <\/mml:mo>\n \n \n \n \u03bc\n \n <\/mml:mi>\n 1<\/mml:mn>\n <\/mml:msub>\n <\/mml:mrow>\n <\/mml:msup>\n ,<\/mml:mo>\n \n \u2026\n \n <\/mml:mo>\n ,<\/mml:mo>\n \n |\n \n <\/mml:mo>\n \n B<\/mml:mi>\n r<\/mml:mi>\n <\/mml:msub>\n \n \n |\n \n <\/mml:mo>\n \n \n \n \u03bc\n \n <\/mml:mi>\n r<\/mml:mi>\n <\/mml:msub>\n <\/mml:mrow>\n <\/mml:msup>\n }<\/mml:mo>\n <\/mml:mrow>\n <\/mml:mrow>\n 1\/{\\max \\{\\lvert B_1 \\rvert ^{\\mu _1},\\ldots ,\\lvert B_r \\rvert ^{\\mu _r}\\}}<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n for all\n \n \n \n \n \n B<\/mml:mi>\n 0<\/mml:mn>\n <\/mml:msub>\n ,<\/mml:mo>\n \n \u2026\n \n <\/mml:mo>\n ,<\/mml:mo>\n \n B<\/mml:mi>\n r<\/mml:mi>\n <\/mml:msub>\n \n \u2208\n \n <\/mml:mo>\n \n \n Z<\/mml:mi>\n <\/mml:mrow>\n \n \n I<\/mml:mi>\n <\/mml:mrow>\n <\/mml:mrow>\n <\/mml:msub>\n <\/mml:mrow>\n B_0,\\ldots ,B_r \\in \\mathbb {Z}_{\\mathbb {I}}<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n ,\n \n \n \n \n max<\/mml:mo>\n {<\/mml:mo>\n \n |\n \n <\/mml:mo>\n \n B<\/mml:mi>\n 1<\/mml:mn>\n <\/mml:msub>\n \n \n |\n \n <\/mml:mo>\n \n \n \n \u03bc\n \n <\/mml:mi>\n 1<\/mml:mn>\n <\/mml:msub>\n <\/mml:mrow>\n <\/mml:msup>\n ,<\/mml:mo>\n \n \u2026\n \n <\/mml:mo>\n ,<\/mml:mo>\n \n |\n \n <\/mml:mo>\n \n B<\/mml:mi>\n r<\/mml:mi>\n <\/mml:msub>\n \n \n |\n \n <\/mml:mo>\n \n \n \n \u03bc\n \n <\/mml:mi>\n r<\/mml:mi>\n <\/mml:msub>\n <\/mml:mrow>\n <\/mml:msup>\n }<\/mml:mo>\n \n \u2265\n \n <\/mml:mo>\n \n H<\/mml:mi>\n 0<\/mml:mn>\n <\/mml:msub>\n <\/mml:mrow>\n \\max \\{\\lvert B_1 \\rvert ^{\\mu _1},\\ldots ,\\lvert B_r \\rvert ^{\\mu _r}\\} \\ge H_0<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n , where the exponents\n \n \n \n \n \n \n \u03bc\n \n <\/mml:mi>\n 1<\/mml:mn>\n <\/mml:msub>\n ,<\/mml:mo>\n \n \u2026\n \n <\/mml:mo>\n ,<\/mml:mo>\n \n \n \u03bc\n \n <\/mml:mi>\n r<\/mml:mi>\n <\/mml:msub>\n <\/mml:mrow>\n \\mu _1,\\ldots ,\\mu _r<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n are different when the given type II approximating sequences approximate some of the numbers\n \n \n \n \n \n \n \u03b8\n \n <\/mml:mi>\n 1<\/mml:mn>\n <\/mml:msub>\n ,<\/mml:mo>\n \n \u2026\n \n <\/mml:mo>\n ,<\/mml:mo>\n \n \n \u03b8\n \n <\/mml:mi>\n r<\/mml:mi>\n <\/mml:msub>\n <\/mml:mrow>\n \\theta _1,\\ldots ,\\theta _r<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n better than the others. As an application we research certain linear forms in logarithms. Our results are completely explicit.\n <\/p>","DOI":"10.1090\/mcom\/3078","type":"journal-article","created":{"date-parts":[[2015,5,27]],"date-time":"2015-05-27T10:44:09Z","timestamp":1432723449000},"page":"2995-3008","source":"Crossref","is-referenced-by-count":0,"title":["Explicit lower bounds for linear forms"],"prefix":"10.1090","volume":"85","author":[{"given":"Kalle","family":"Lepp\u00e4l\u00e4","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"14","published-online":{"date-parts":[[2016,1,19]]},"reference":[{"key":"1","doi-asserted-by":"publisher","first-page":"616","DOI":"10.4153\/CJM-1965-061-8","article-title":"On some Diophantine inequalities involving the exponential function","volume":"17","author":"Baker, A.","year":"1965","journal-title":"Canadian J. 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