Inhomogeneous approximation with coprime integers and lattice orbits (Q2895961)

\textit{J. H. H. Chalk} and \textit{P. Erdős} [Can. Math. Bull. 2, 91--96 (1959; Zbl 0088.25703)] proved: Let \(\xi\) be an irrational real number and let \(y\) be a real number. There exists an absolute constant \(c\) such that the inequality NEWLINE\[NEWLINE|q\xi+p-y|\leq \frac{c(\log\;q)^2}{q(\log \log q)^2},\tag{e1}NEWLINE\]NEWLINE holds for infinitely many pairs of coprime integers \((p,q)\) with \(q>0\). NEWLINENEWLINEThe authors study more generally the Diophantine inequality \(|q\xi+p-y|\leq \psi(|q|)\) for coprime integers \(p\) and \(q\) where \(\psi: \mathbb N\rightarrow \mathbb R^+\) is a given function. At first, they prove:NEWLINENEWLINELet \(\xi\) be an irrational real number and let \(y\) be a nonzero real number. There exist infinitely many integer quadruples \((p_1,q_1,p_2,q_2)\) satisfying \(q_1p_2-p_1q_2=1\) and NEWLINE\[NEWLINE|q_i\xi+p_i-y|\leq \frac{c}{\max (|q_1|,|q_2|)^{1/2}}\leq \frac{c}{\sqrt{|q_i|}}\;(i=1,2)NEWLINE\]NEWLINE with NEWLINE\[NEWLINEc=2\sqrt{3}\max(|1,|\xi|^{1/2}|y|^{1/2}).NEWLINE\]NEWLINE NEWLINEThen, they adress the following problem: Can we replace the function \(\psi(\ell)=\frac{c(\log \ell)^2}{\ell(\log\log \ell)^2}\) occurring in (e1) by a smaller one, possibly \(\psi(\ell)= c\ell^{-1}\)? Towards this problem they prove the following metrical statement.NEWLINENEWLINENEWLINETheorem. Let \(\psi: \mathbb N\to\mathbb R^+\) be a function. Assume that \(\psi\) is non-increasing, tends to \(0\) at infinity and that, for every positive integer \(c\), there exists a positive real number \(c_1\) satisfying \(\psi(c\ell)\geq c_1\psi(\ell), \;\forall \ell\geq1\). Furthermore assume that \(\sum_{\ell\geq 1} \psi(\ell)=+\infty\). Then, for almost all pairs \((\xi,y)\) of real numbers, there exist infinitely many \textit{primitive points} \((p,q)\) (so with \(\gcd(p,q)=1\)) such that NEWLINE\[NEWLINEq\geq 1\text{\;and\;}|q\xi+p-y|\leq \psi(q).\tag{e4}NEWLINE\]NEWLINE If \(\sum_{\ell\geq 1} \psi(\ell)\) converges, the pairs \((\xi, y)\) satisfying (e4) for infinitely many primitive points \((p,q)\) form a set of zero Lebesgue measure.NEWLINENEWLINENEWLINEIn the following result they fix now \(\xi\) to get:NEWLINENEWLINENEWLINETheorem. Let \(\xi\) be an irrational number and let \((p_k/q_k)_{k\geq 0}\) be the sequence of its convergents. Assume that the series NEWLINE\[NEWLINE \sum_{k\geq 0}\frac{1}{\max( 1, \log q_k)}\tag{e5}NEWLINE\]NEWLINE diverges. Then for almost every real number \(y\) there exist infinitely many primitive points \((p,q)\) satisfying \(|q\xi+p-y|\leq\frac{2}{|q|}\). Moreover the series (e5) diverges for almost every real \(\xi\).NEWLINENEWLINEThe second part of the article is devoted, in the more general setting of lattices \(\Gamma\) in \(\mathrm{SL}(2,\mathbb R)\) and \textit{ergodic theory} (see \textit{C. Moore} [Am. J. Math. 88, 154--178 (1966; Zbl 0148.37902)]), to evaluation of \textit{generic density exponents} for lattices orbits \(\Gamma\mathbf x\) in \(\mathbb R^2\). The authors intend to show that the exponent \(1/2\) is best possible in general. See also \textit{V. Beresnevich} et al. [Analytic number theory. Essays in honour of Klaus Roth on the occasion of his 80th birthday. Cambridge: Cambridge University Press, 38--61 (2009; Zbl 1236.11064)]; [\textit{G. Harman}, Metric number theory. Oxford: Clarendon Press (1998; Zbl 1081.11057)] on the \textit{metric number theory} aspects in connection with this paper.