2024/01/16 by Volodymyr V. Pavlenkov, Pavlenkov, Volodymyr, Evgeniy Zorin +1
Computer Science · Mathematics · #11J83 (Primary) #42A61 (Secondary) #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals #Number Theory (math.NT) #Topological and Geometric Data Analysis #advanced mathematical theories
paper · pdf · doi:10.48550/arxiv.2401.08849
openalex publication_date 2024/01/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We prove new quantitative Schmidt-type theorem for Diophantine approximations with restraint denominators on fractals (more precisely, on M0-sets). Our theorems introduce a sharp balance condition between the growth rate of the sequence of denominators and the decay rate of the Fourier transform of a Rajchman measure. Among the other things, this allows applications to sequences of denominators of polynomial growth. In particular, we infer new inhomogeneous Khintchine-Järnik type theorems with restraint denominators for a broad family of denominator sequences. Furthermore, our results provide non-trivial lower bounds for Hausdorff dimensions of intersections of two sets of inhomogeneously well-approximable numbers with restraint denominators.