Metrical properties of weighted products of consecutive Lüroth digits (Q6589527)

The present research deals with the Lebesgue measure and the Hausdorff dimension for certain sets whose elements defined in terms of Lüroth series.\N\NSeries of the form \N\[\N\frac{1}{d_1} + \frac{1}{d_1 (d_1 - 1) d_2} + \frac{1}{d_1 (d_1 - 1) d_2 (d_2 - 1) d_3} + \cdots,\N\]\Nwhere \(d_k\) is a positive integer number and \(d_k>1\) for all \(k=1, 2, 3, \dots\), are called Lüroth series. Any number \(x \in (0, 1]\) can be represented by these series.\N\NGiven \(m \in \mathbb{N}\), \(\mathbf{t} = (t_0, \dots, t_{m-1}) \in \mathbb{R}_{>0}^{m-1}\), and \(\Psi : \mathbb{N} \to (1, \infty)\) any function.\N\NThe main attention is given a set of the form: \N\[\N\mathcal{E}_{\mathbf{t}} (\Psi) := \Big\{x \in (0, 1] : d^{t_0}_n \cdots d^{t_{m-1}}_{n+m} \geq \Psi (n) \text{ for infinitely many } n \in \mathbb{N} \Big\}.\N\]\NThe main results are the following:\N\N-- To establish ``a Lebesgue measure dichotomy statement (a zero-one law) for \(\mathcal{E}_{\mathbf{t}} (\Psi)\) under a natural non-removable condition\( \liminf_{n \to \infty} \Psi (n) > 1\)''.\N\N-- To calculate the Hausdorff dimension of \(\mathcal{E}_{\mathbf{t}} (\Psi)\) for any \(m\in \mathbb N\) for the cases when either \(B=1\) or \(B=\infty\), as well as for the case when \(1<B<\infty\) for \(m=2\). Here \N\[\N\log B := \liminf_{n \to \infty} \frac{\log(\Psi (n))}{n}.\N\]\NA brief survey of this paper is devoted to the metric theory of Lüroth series. The Borel-Bernstein Theorem for Lüroth series and some properties of Lüroth series are described.\N\NSeveral auxiliary statements are proven. All proofs are given with explanations. Finally, one conjecture is discussed.