Note on a Diophantine Approximation

Multiple ergodic averages along functions from a Hardy field: convergence, recurrence and combinatorial applications

“…which, for C sufficiently small, is an infinite set of zero upper Banach density by the classical Tchebychef's inhomogenous version of Dirichlet's Approximation Theorem (see [Gra18]).…”

Inhomogeneous Diophantine approximation in the coprime setting

“…Grace [10] showed that 1/4 in (2) is sharp, and Khintchine [6] showed that lim inf |q|→∞ |q| ||qα − γ|| ≤ 1 4 (1 − 4λ(α) 2 ) Uniform inhomogeneous coprime approximation was studied by Chalk and Erdós who proved [5] that for any irrational α ∈ R and for any γ there are infinitely many pairs of coprime integers (p, q) such that (1) holds with ψ(q) = ( log q log log q ) 2 1 q . Laurent and Nogueira [15] conjectured that a result similar to Minkowski's theorem holds also for the inhomogeneous coprime approximation, namely that there exists C > 0 such that for any for any irrational α ∈ R and for any γ there are infinitely many pairs of coprime integers (p, q) with…”

Inhomogeneous Diophantine approximation of some Hurwitzian numbers

“…Theorem 2.1 was implicit in Grace's work [8]. We illustrate its usefulness by proving that for any integer k ≥ 3, L(e 2/k , (e 2/k + 1)/2) = 0.…”

“…In particular, Komatsu used several different types of continued fractions to compute the inhomogeneous constants when e 1/s (for positive integer s) is paired with various φ in Q θ + Q. In this article we make use of the "relative rationality" of these pairs θ, φ to show how the technically simpler ideas of Grace [8] and regular simple continued fractions can be used to unify and extend Komatsu's results.…”

“…Minkowski proved that when θ is irrational, L(θ, φ) ≤ 1/4 holds for all φ. Grace [8] used regular simple continued fractions to construct θ with L(θ, 1/2) = 1/4. Further historical details on these and related results can be found in Koksma [9].…”