## 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
$\inf_{q\in\mathbb N^*} q\|q\alpha\|\,|q|_{\mathcal B}=0 \tag{1}$
holds.
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 $$\sup_{\mathcal S} \lambda({\mathcal S})<+\infty$$.

### MSC:

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

Zbl 1162.11361
Full Text:

### References:

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