2 results
SOME REFINED RESULTS ON THE MIXED LITTLEWOOD CONJECTURE FOR PSEUDO-ABSOLUTE VALUES
- Part of
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- Journal:
- Journal of the Australian Mathematical Society / Volume 107 / Issue 1 / August 2019
- Published online by Cambridge University Press:
- 22 August 2018, pp. 91-109
- Print publication:
- August 2019
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- Article
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In this paper, we study the mixed Littlewood conjecture with pseudo-absolute values. For any pseudo-absolute-value sequence
${\mathcal{D}}$, we obtain a sharp criterion such that for almost every 
$\unicode[STIX]{x1D6FC}$ the inequality has infinitely many coprime solutions
$$\begin{eqnarray}|n|_{{\mathcal{D}}}|n\unicode[STIX]{x1D6FC}-p|\leq \unicode[STIX]{x1D713}(n)\end{eqnarray}$$
$(n,p)\in \mathbb{N}\times \mathbb{Z}$ for a certain one-parameter family of 
$\unicode[STIX]{x1D713}$. Also, under a minor condition on pseudo-absolute-value sequences 
${\mathcal{D}}_{1},{\mathcal{D}}_{2},\ldots ,{\mathcal{D}}_{k}$, we obtain a sharp criterion on a general sequence 
$\unicode[STIX]{x1D713}(n)$ such that for almost every 
$\unicode[STIX]{x1D6FC}$ the inequality has infinitely many coprime solutions
$$\begin{eqnarray}|n|_{{\mathcal{D}}_{1}}|n|_{{\mathcal{D}}_{2}}\cdots |n|_{{\mathcal{D}}_{k}}|n\unicode[STIX]{x1D6FC}-p|\leq \unicode[STIX]{x1D713}(n)\end{eqnarray}$$
$(n,p)\in \mathbb{N}\times \mathbb{Z}$.
On the complexity of a putative counterexample to the
$p$-adic Littlewood conjecture
- Part of
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- Journal:
- Compositio Mathematica / Volume 151 / Issue 9 / September 2015
- Published online by Cambridge University Press:
- 19 May 2015, pp. 1647-1662
- Print publication:
- September 2015
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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.
