Bounding the largest inhomogeneous Diophantine approximation constant
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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.