On a mixed Littlewood conjecture for quadratic numbers. (English) Zbl 1165.11325

From the introduction: In a joint paper, with O. Teulié [Monatsh. Math. 143, No. 3, 229–245 (2004; Zbl 1162.11361)], we have considered the following problem. Let \({\mathcal B}= (b_k)_{k\geq1}\) be a sequence of integers greater than 1. Consider the sequence \((r_n)_{n\geq0}\), where \(r_0=1\) and \(r_n= \prod_{0<k\leq n}b_k\) for \(n>0\). For \(q\in\mathbb Z\), set \(w_{\mathcal B}(q)= \sup\{n\in\mathbb N\); \(q\in r_n\mathbb Z\}\) and \(|q|_{\mathcal B}= \inf\{1/r_n\); \(q\in r_n\mathbb Z\}\).
Note that \(|.|_{\mathcal B}\) is not necessarily an absolute value, but when \({\mathcal B}\) is the constant sequence \(p\), where \(p\) is a prime number, then \(|.|_{\mathcal B}\) is the usual \(p\)-adic value. For \(x\in\mathbb R\), we denote by \(\{x\}\) the number in \([-1/2,1/2[\) such that \(x-\{x\}\in\mathbb Z\). As usual, we put \(\|x\|= |\{x\}|\).
Let \(\alpha\) be a real number. Given a positive integer \(M\), Dirichlet’s theorem asserts that for any \(n\), there exists an integer \(q\), with \(0<q\leq Mr_n\), satisfying simultaneously the approximation condition \(\|q\|< 1/M\) and the divisibility condition \(r_n|q\), i.e., \(|q|_{\mathcal B}\leq 1/r_n\). Indeed, it is enough to apply Dirichlet’s theorem to the number \(r_n\alpha\). We thus find positive integers \(q\) with
\[ q\|q\alpha\|\,|q|_{\mathcal B}<1. \]
By analogy with Littlewood’s conjecture, we ask whether
\[ \inf_{q\in\mathbb N^*} q\|q\alpha\|\,|q|_{\mathcal B}=0 \tag{1} \]
We do not know whether (1) is satisfied for any real number \(\alpha\). In (loc. cit.), we have proved that if we assume that the sequence \({\mathcal B}= (b_k)_{k\geq1}\) is bounded, (1) is true for every quadratic number \(\alpha\). Here we prove:
Theorem. Assume that the sequence \({\mathcal B}\) is bounded. Let \(\alpha\) be a real quadratic number, and let \({\mathcal S}\) be a set of integers \(q>1\) with
\[ \|q\alpha\|\ll 1/q.\tag{2} \]
Then there exists a constant \(\lambda= \lambda({\mathcal S})\) such that
\[ |q|_{\mathcal B}\gg \frac{1}{(\ln q)^\lambda}, \tag{4} \]
for any \(q\in{\mathcal S}\).
One may expect that (4) holds for any \(\lambda>1\), but we are not able to prove this. We do not even know whether there exists a real number \(\lambda\) for which (4) holds for any set \({\mathcal S}\) of integers \(q>1\) satisfying (2). Indeed, Theorem 1.2 does not ensure that \(\sup_{\mathcal S} \lambda({\mathcal S})<+\infty\).


11J13 Simultaneous homogeneous approximation, linear forms
11J61 Approximation in non-Archimedean valuations


Zbl 1162.11361
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