2023/01/20 by Bishnu Hari Paudel, Paudel, Bishnu, Chris Pinner +1
Computer Science · Mathematics · #11J06 #11J20 #11J70 #FOS: Mathematics #Mathematical Approximation and Integration #Mathematical functions and polynomials #Number Theory (math.NT) #Numerical Methods and Algorithms
paper · pdf · doi:10.48550/arxiv.2301.08825
openalex publication_date 2023/01/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
For a given irrational number α and a real number γ in (0,1) one defines the two-sided inhomogeneous approximation constant M(α,γ):=\liminf|n|→∞|n| ||nα-γ||, and the case of worst inhomogeneous approximation for α ρ(α):=supγ∉ℤ+αℤM(α,γ). We are interested in lower bounds on ρ(α) in terms of R:=\liminfi→∞ai, where the ai are the partial quotients in the negative (i.e. the `round-up') continued fraction expansion of α. We obtain bounds for any R≥ 3 which are best possible when R is even (and asymptotically precise when R is odd). In particular when R≥ 3 ρ(α)≥ \cfrac16√(3)+8=\cfrac118.3923…, and when R≥ 4, optimally, ρ(α) ≥ \cfrac14√(3)+2=\cfrac18.9282….