Lower Bounds on the Largest Inhomogeneous Approximation Constant

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):=lim| n |→∞ inf | n |‖ nα-γ ‖, M\left( {\alpha ,\gamma } \right): = \mathop {\lim }\limits_{\left| n \right| \to \infty } \,\,\inf \,\,\,\left| n \right|\left\| {n\alpha - \gamma } \right\|, and the case of worst inhomogeneous approximation for α ρ(α):=supγ∉{ m+lα:m,l∈ℤ }M(α,γ). \rho \left( \alpha \right): = \mathop {\sup }\limits_{\gamma \notin \left\{ {m + l\alpha :m,l \in \mathbb Z} \right\}} M\left( {\alpha ,\gamma } \right).

We are interested in lower bounds on ρ(α)in terms of R := lim inf_i→∞ a_i, where the a_i 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)≥163+8=118.3923…, \rho \left( \alpha \right) \ge {1 \over {6\sqrt 3 + 8}} = {1 \over {18.3923 \ldots }}, and when R ≥ 4, optimally, ρ(α)≥143+2=18.9282…, \rho \left( \alpha \right) \ge {1 \over {4\sqrt 3 + 2}} = {1 \over {8.9282 \ldots }},