Hausdorff dimension and distance sets. (English) Zbl 0807.28004
The author uses a brilliant refinement of Fourier restriction phenomena to spheres to improve results on difference sets \(D(A)= \{| x- y|; x,y\in A\}\) for Souslin sets \(A\) in \(\mathbb{R}^ n\) due to Falconer. For example, if \(A\subset \mathbb{R}^ 2\) and the Hausdorff dimension \(\dim A> {13\over 9}\) (instead of \({3\over 2}\) as in Falconer’s general result) then \(D(A)\) has positive measure.
Reviewer: H.Haase (Greifswald)
MSC:
| 28A78 | Hausdorff and packing measures |
| 43A60 | Almost periodic functions on groups and semigroups and their generalizations (recurrent functions, distal functions, etc.); almost automorphic functions |
References:
| [1] | Bourgain, J., Besicovitch type maximal functions and applications to Fourier analysis, Geometric and Functional Analysis, 1, 147-187 (1991) · Zbl 0756.42014 · doi:10.1007/BF01896376 |
| [2] | Bourgain, J., On the restriction and multiplier problems in ℝ^3, 179-191 (1991), Berlin: Springer-Verlag, Berlin · Zbl 0792.42004 |
| [3] | Falconer, K., On the Hausdorff dimension of distance sets, Mathematika, 32, 206-212 (1985) · Zbl 0605.28005 |
| [4] | K. Falconer,The Geometry of Fractal Sets, Cambridge Tracts in Math. 85, Cambridge University Press, 1984. |
| [5] | Mattila, P., Spherical averages of Fourier transforms of measures with finite energy; dimension of intersections and distance sets, Mathematika, 34, 207-228 (1987) · Zbl 0645.28004 · doi:10.1112/S0025579300013462 |
| [6] | E. Stein,Beijing Lectures in harmonic Analysis, No. 112, Princeton University Press, 1986, pp. 305-355. · Zbl 0595.00015 |
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