Annales Academiæ Scientiarum Fennicæ
Mathematica
Volumen 42, 2017, 611-620

EXCEPTIONAL SET ESTIMATES FOR THE HAUSDORFF DIMENSION OF INTERSECTIONS

Pertti Mattila

University of Helsinki, Department of Mathematics and Statistics
P.O. Box 68, FI-00014 University of Helsinki, Finland; pertti.mattila 'at' helsinki.fi

Abstract. Let A and B be Borel subsets of the Euclidean n-space with dim A + dim B > n and let 0 < u < dim A + dim B – n where dim denotes Hausdorff dimension. Let E be the set of those orthogonal transformations g ∈ O(n) for which dim A ∩ (g(B) + z) ≤ u for almost all z ∈ Rⁿ. If dim A + dim B > n + 1, then dim E ≤ n(n – 1)/2 + 1 \ndash; u, and if dim A ≤ (n – 1)/2, then dim E ≤ n(n – 1)/2 – u. If A is a Salem set and 0 < u < dim A + dim B – n and dim A + dim B > 2n – 1, then dim A ∩ (B + z) > u for z in a set of positive Lebesgue measure. If dim A + dim B ≤ 2n – 1, the set of exceptional g ∈ O(n) has dimension at most n(n – 1)/2 – u.

2010 Mathematics Subject Classification: Primary 28A75.

Key words: Hausdorff dimension, intersection, energy integral, Fourier transform.

Reference to this article: P. Mattila: Exceptional set estimates for the Hausdorff dimension of intersections. Ann. Acad. Sci. Fenn. Math. 42 (2017), 611-620.

Full document as PDF file

https://doi.org/10.5186/aasfm.2017.4236