Fractal Intersections and Products via Algorithmic Dimension

Algorithmic fractal dimensions quantify the algorithmic information density of individual points and may be defined in terms of Kolmogorov complexity. This work uses these dimensions to bound the classical Hausdorff and packing dimensions of intersections and Cartesian products of fractals in Euclidean spaces. This approach shows that two prominent, fundamental results about the dimension of Borel or analytic sets also hold for arbitrary sets.

💡 Research Summary

The paper investigates the relationship between algorithmic information theory and classical fractal geometry by employing algorithmic dimensions—namely, the lower and upper algorithmic dimensions dim(x) and Dim(x) defined via Kolmogorov complexity—to bound Hausdorff and packing dimensions of arbitrary subsets of Euclidean spaces. The authors build on the point‑to‑set principle introduced by Lutz and Lutz, which states that the Hausdorff dimension of a set E equals the minimum over all oracles A of the supremum of dim_A(x) for x in E, and similarly the packing dimension equals the supremum of Dim_A(x). This principle enables a pointwise approach to dimension lower bounds.

The first major result (Theorem 1) extends a classical intersection formula, previously known only for Borel sets, to all subsets E, F ⊆ ℝⁿ. It shows that for Lebesgue‑almost every translation vector z, the Hausdorff dimension of the intersection satisfies
 dim_H(E ∩ (F+z)) ≤ max{0, dim_H(E×F) − n}.
The proof selects a Hausdorff oracle A for the product E×F, finds a point x in the intersection with high conditional dimension relative to z, and uses the chain rule for dimensions together with Lemma 6 (which relates relative and conditional dimensions) to derive the inequality. Since a random point z relative to A has full dimension n, the inequality holds for almost all z.

A symmetric argument yields the analogous packing‑dimension statement (Theorem 9): for almost all z,
 dim_P(E ∩ (F+z)) ≤ max{0, dim_P(E×F) − n}.
Both results rely only on Lemma 4 (invariance under computable bi‑Lipschitz maps), Theorem 5 (the chain rule for conditional dimensions), Lemma 6, and the point‑to‑set principle; explicit Kolmogorov complexity calculations are not needed.

The second major contribution (Theorem 2) provides a new characterization of packing dimension for arbitrary sets:
 dim_P(E) = sup_{F⊆ℝⁿ}