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We prove that the upper Minkowski dimension of a compact set Λ is equal to the convergence exponent of any packing of the complement of Λ with polyhedra of size not smaller than a constant multiple of their distance from Λ.

For a linear solenoid with two different contraction coefficients and box dimension greater than 2, we give precise formulas for the Hausdorff and packing dimensions. We prove that the packing measure is infinite and give a condition necessary and sufficient for the Hausdorff measure to be positive, finite and equivalent to the SBR measure. We also give analogous results, generalizing [P], for affine IFS in ℝ².

We estimate from above and below the Hausdorff dimension of SRB measure for contracting-on-average baker maps.

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.

We consider the packing spectra for the local dimension of Bernoulli measures supported on Bedford-McMullen carpets. We show that typically the packing dimension of the regular set is smaller than the packing dimension of the attractor. We also consider a specific class of measures for which we are able to calculate the packing spectrum exactly, and we show that the packing spectrum is discontinuous as a function on the space of Bernoulli measures.