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Let α be an irrational and φ: ℕ → ℝ⁺ be a function decreasing to zero. Let $\omega \left(\alpha \right):=sup\theta \ge 1:limin{f}_{n\to \infty }{n}^{\theta }\left|\right|n\alpha |=0$ $.Forany\alpha withagiven\omega \left(\alpha \right),wegivesomesharpestimatesfortheHausdorffdimensionoftheset$ $E_{\phi }\left(\alpha \right)$ := y ∈ ℝ: ||nα -y|| < φ(n) for infinitely many n, where ||·|| denotes the distance to the nearest integer.