EXPLICIT LOWER BOUNDS FOR LINEAR FORMS

Publication: Research - peer-reviewJournal article

DOI

Let I be the field of rational numbers or an imaginary quadratic field and Z(I) its ring of integers. We study some general lemmas that produce lower bounds

vertical bar B-0 + B-1 theta(1) +... + B-r theta(r)vertical bar >= 1/max{vertical bar B-1 vertical bar,...,vertical bar B-r vertical bar}(mu)

for all B-0,...,B-r is an element of Z(I), max{vertical bar B-1 vertical bar,...,vertical bar B-r vertical bar} >= H-0, given suitable simultaneous approximating sequences of the numbers theta(1),...,theta(r). We manage to replace the lower bound with 1/max{vertical bar B-1 vertical bar(mu 1),...,vertical bar Br vertical bar(mu r)} for all B-0,..., B-r is an element of Z(I), max{vertical bar B-1 vertical bar(mu 1),...,vertical bar Br vertical bar(mu r)} >= H-0, where the exponents mu(1),...,mu(r) are different when the given type II approximating sequences approximate some of the numbers theta(1),...,theta(r) better than the others. As an application we research certain linear forms in logarithms. Our results are completely explicit.

Original languageEnglish
JournalMathematics of Computation
Volume85
Issue number302
Pages (from-to)2995-3008
Number of pages14
ISSN0025-5718
DOIs
StatePublished - Nov 2016

    Keywords

  • IRRATIONALITY MEASURE, LOGARITHMS, NUMBERS, PI

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