2022/03/19 by Sam Chow, Chow, Sam, Niclas Technau +1 · 1 citation
Mathematics · #Mathematical Dynamics and Fractals #Analytic Number Theory Research #Mathematical Approximation and Integration
paper · pdf · doi:10.48550/arxiv.2203.10380
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, Wang and Yu (1981) 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.