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The mixed Littlewood conjecture for pseudo-absolute values. (English) Zbl 1300.11082

In this paper the authors investigate problems related to a modified version of the well-known Littlewood conjecture, which is known as the mixed Littlewood conjecture or as the de Mathan–Teulié conjecture. This conjecture asserts that for every \(\alpha \in \mathbb{R}\) \[ \inf_{n \in \mathbb{N}} n |n|_{\mathcal{D}} \| n \alpha\| = 0, \] where \(\| \cdot \|\) denotes the distance to the nearest integer and \(|\cdot|_{\mathcal{D}}\) is a so-called (\(\mathcal{D}\)-adic) pseudo-absolute value which is defined by \[ |n|_{\mathcal{D}} = \min\left\{n_k^{-1}:~n \in n_k \mathbb{Z}\right\} \] for \(\mathcal{D} = (n_k)_{k \geq 0}\) being an increasing sequence of integers with \(n_0=1\) and \(n_k | n_{k+1}\) for all \(k\). (If \(\mathcal{D}=(a^k)_{k \geq 0}\) for some integer \(a \geq 2\), then we write \(|\cdot|_a = |\cdot|_{\mathcal{D}}\); with this notation, if \(p\) is a prime, then \(|\cdot|_p\) is the usual \(p\)-adic absolute value.)
The main theorem of the paper under review is the following:
{ Theorem 1.} Let \(a \geq 2\) be an integer, and let \(\mathcal{D} = (n_k)_{k \geq 1}\) be a pseudo-absolute value sequence such that all elements of \(\mathcal{D}\) are divisible by finitely many fixed primes which are coprime to \(a\). If there exists an \(R \geq 0\) such that \[ \log n_k \leq k^R, \qquad k \geq 2, \] then for all \(\alpha \in \mathbb{R}\) \[ \inf_{n \in \mathbb{N}} n |n|_a |n|_\mathcal{D} \|n \alpha \| = 0. \]
The proof combines methods from the theory of dynamical systems (H. Furstenberg’s orbit closure theorem [Math. Syst. Theory 1, 1–49 (1967; Zbl 0146.28502)], results on measure rigidity by E. Lindenstrauss [Ann. Math. (2) 163, No. 1, 165–219 (2006; Zbl 1104.22015)] and others) with methods from Diophantine approximation (Baker’s lower bounds for linear forms in logarithms).
A second theorem characterizes those sequences \(\psi(n)\) for which the inequality \[ |n|_\mathcal{D} \|n \alpha\| \leq \psi(n) \] has infinitely many solutions \(n \in \mathbb{N}\) for almost all \(\alpha\).

MSC:

11K60 Diophantine approximation in probabilistic number theory
37A45 Relations of ergodic theory with number theory and harmonic analysis (MSC2010)
11J83 Metric theory
11J86 Linear forms in logarithms; Baker’s method
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References:

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