Real Analysis Exchange

Dimensions of intersections and distance sets for polyhedral norms.

K. J. Falconer

Abstract

We obtain an estimate for the typical Hausdorff dimension of the intersection of a set $E$ with homothetic copies of a set $F$, where $E$ and $F$ are Borel subsets of $\mathbb{R}^{n}$. We apply this to the `distance set problem' for a polyhedral norm on $\mathbb{R}^{n}$, by showing that there are subsets of full dimension with distance set of Lebesgue measure 0.

Article information

Source
Real Anal. Exchange Volume 30, Number 2 (2004), 719- 726 .

Dates
First available in Project Euclid: 15 October 2005

http://projecteuclid.org/euclid.rae/1129416466

Mathematical Reviews number (MathSciNet)
MR2177429

Zentralblatt MATH identifier
1107.28008

Citation

Falconer, K. J. Dimensions of intersections and distance sets for polyhedral norms. Real Anal. Exchange 30 (2004), no. 2, 719-- 726. http://projecteuclid.org/euclid.rae/1129416466.

References

• J. Bougain, Hausdorff dimension and distance sets, Israel J. Math., 87 (1994), 193–201.
• M. B. Erdogan, On Falconer's distance set conjecture, Rev. Mat. Ibero, to appear.
• K. J. Falconer, On the Hausdorff dimension of distance sets, Mathematika, 32 (1986), 206–212.
• K. J. Falconer, Fractal Geometry - Mathematical Foundations and Applications, 2nd ed., John Wiley, 2003.
• A. Iosevich and I. Laba, $K$-distance sets, Falconer conjecture and discrete analogs, Integers: Electronic Journal of Combinatorial Number Theory, to appear.
• J.-P. Kahane, Sur la dimension des intersections, Aspects of Mathematics and its Applications, ed. J.A. Barroso, Elsevier Science Publishers, 1986, 419–430.
• S. Konyagin and I. Laba, Distance sets of well distributed planar sets for polygonal norms, Israel J. Math., to appear.
• S. Konyagin and I. Laba, Separated sets and the Falconer conjecture for polygonal norms, preprint 2004.
• J. M. Marstrand, Some fundamental properties of plane sets of fractional dimensions, Proc. London Math. Soc., (3) 4 (1954), 257–302.
• P. Mattila, Hausdorff dimension, orthogonal projections and intersections with planes, Ann. Acad. Sci. Fenn. Ser. A, 4 (1978/79), 53–61.
• P. Mattila, Hausdorff dimension and capacities of intersections of sets in $n$-space, Acta Math., 152 (1984), 77–105.
• P. Mattila, Hausdorff dimension and capacities of intersections, Mathematika, 32 (1985), 213–217.
• P. Mattila, Geometry of Sets and Measures in Euclidean Spaces, Cambridge University Press, 1995.
• T. Wolff, Decay of circular means of Fourier transforms of measures, Int. Math. Res. Notices, 10 (1999), 546–567.