Bounding the largest inhomogeneous Diophantine approximation constant

Abstract

For an irrational real α and real γ ∉ αZ + Z, one defines the two-sided inhomogeneous approximation constant M(α,γ):= liminf_{|n|→∞} |n| ||nα-γ||, and the worst-case of inhomogeneous approximation ρ(α):=sup_{ γ ∉ αZ + Z} M(α,γ). By a well-known theorem of Minkowski, we have ρ(α) ≤ 1/4. This dissertation focuses on bounding ρ(α) in terms of R:=liminf_{i→∞} aᵢ, where aᵢ are the partial quotients in the negative (i.e. the `round-up') continued fraction expansion α. We prove that if R is odd, then the upper bound 1/4 can be replaced by 1/4(1-1/R)(1-1/R²), which is optimal. The optimal upper bound for even R≥ 4 was already known.

We also obtain bounds of the form ρ(α)≥ C(R) for any R≥ 3 which are best possible when R is even (and asymptotically precise when R is odd). In particular, ρ(α) ≥ 1/(6√3+8)=1/18.3923..., when R=3, 1/(4√3+2)=1/8.9282…, when R≥4.