2025/06/04 by Michaud, Gilbert, Felipe A. Ramírez, Ramírez, Felipe A. · 1 citation
Mathematics · Physics and Astronomy · #11J83 #11K60 #Advanced Differential Equations and Dynamical Systems #FOS: Mathematics #Mathematical Dynamics and Fractals #Number Theory (math.NT) #Quantum chaos and dynamical systems
paper · pdf · doi:10.48550/arxiv.2506.04187
openalex publication_date 2025/06/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Szüsz's inhomogeneous version (1958) of Khintchine's theorem (1924) gives conditions on ψ:ℕ→ℝ≥ 0 under which for almost every real number α there exist infinitely many rationals p/q such that |α- (p+γ)/(q)| lt; (ψ(q))/(q), where γ∈ℝ is some fixed inhomogeneous parameter. It is often interpreted as a statement about visits of qα (\bmod 1) to a shrinking target centered around γ (\bmod 1), viewed in ℝ/ℤ. Hauke and the second author have conjectured that Szüsz's result continues to hold if the target is allowed to move as well as shrink, that is, if the inhomogeneous parameter γ is allowed to depend on the denominator q of the approximating rational. We show that the conjecture holds under an ``extra divergence'' assumption on ψ. We also show that it holds when the inhomogeneous parameter's movement is constrained to a finite set. As a byproduct, we obtain a finite-colorings version of the inhomogeneous Khintchine theorem, giving rational approximations with monochromatic denominators.