Counting multiplicative approximations (Q6073936)

This paper deals with certain refinements for one problem, which is related to the Littlewood conjecture and theorems of Khintchine, Gallagher, and Schmidt, as well as Wang and Yu, according to the popularisation by Beresnevich-Haynes-Velani. Let us describe authors' abstract: ``A famous conjecture of Littlewood (c. 1930) concerns approximating two real numbers by rationals of the same denominator, multiplying the errors. In a lesser-known paper, \textit{Y. Wang} and \textit{K. Yu}, Chin Ann Math 2, 1--12 (1981; Zbl 0464.10039)] established an asymptotic formula for the number of such approximations, valid almost always. Using the quantitative Koukoulopoulos-Maynard theorem of Aistleitner-Borda-Hauke, together with bounds arising from the theory of Bohr sets, we deduce lower bounds of the expected order of magnitude for inhomogeneous and fibre refinements of the problem.'' One can note that the special attention is given to a certain system of related results, auxiliary notions, and to explanations of techniques for proving. For example, here the foundational result of metric Diophantine approximations and one its generalization, which is related to the Littlewood conjecture, are discussed.