On the complexity of a putative counterexample to the -adic Littlewood conjecture
Let $\Vert \cdot \Vert$ denote the distance to the nearest integer and, for a prime number $p$ , let $|\cdot |_{p}$ denote the $p$ -adic absolute value. Over a decade ago, de Mathan and Teulié [ Problèmes diophantiens simultanés , Monatsh. Math. 143 (2004), 229–245] asked whether $\inf _{q\geqslant...
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Published in | Compositio mathematica Vol. 151; no. 9; pp. 1647 - 1662 |
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Main Authors | , , , |
Format | Journal Article |
Language | English |
Published |
Foundation Compositio Mathematica
01.09.2015
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Subjects | |
Online Access | Get full text |
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Summary: | Let
$\Vert \cdot \Vert$
denote the distance to the nearest integer and, for a prime number
$p$
, let
$|\cdot |_{p}$
denote the
$p$
-adic absolute value. Over a decade ago, de Mathan and Teulié [
Problèmes diophantiens simultanés
, Monatsh. Math.
143
(2004), 229–245] asked whether
$\inf _{q\geqslant 1}$
$q\cdot \Vert q{\it\alpha}\Vert \cdot |q|_{p}=0$
holds for every badly approximable real number
${\it\alpha}$
and every prime number
$p$
. Among other results, we establish that, if the complexity of the sequence of partial quotients of a real number
${\it\alpha}$
grows too rapidly or too slowly, then their conjecture is true for the pair
$({\it\alpha},p)$
with
$p$
an arbitrary prime. |
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Bibliography: | ObjectType-Article-1 SourceType-Scholarly Journals-1 ObjectType-Feature-2 content type line 23 |
ISSN: | 0010-437X 1570-5846 |
DOI: | 10.1112/S0010437X15007393 |