On the Hausdorff dimensions of distance sets. (English) Zbl 0605.28005
The distance set of a subset E of \(R^ n\) is \(D(E)=\{| x- y|:x,y\in E\}.\) If E is analytic (i.e. Suslin), the author uses Fourier transform to derive the following lower bound for the Hausdorff dimension of \(E\):
\[
\dim D(E)\geq \min \{1,(\dim E)-(n-1)/2\}.
\]
Moreover, \(D(E)\) has positive Lebesgue measure if \(\dim E>(n+1)/2\). The continuum hypothesis is used to show that for general non-analytic sets no such results hold.
Reviewer: P.Mattila
MSC:
| 28A75 | Length, area, volume, other geometric measure theory |
References:
| [1] | Davies, Colloq. Math 42 pp 53– (1979) |
| [2] | Davies, Indag. Math 14 pp 488– (1952) · doi:10.1016/S1385-7258(52)50068-4 |
| [3] | DOI: 10.1016/0097-3165(84)90041-4 · Zbl 0536.05003 · doi:10.1016/0097-3165(84)90041-4 |
| [4] | Carleson, Selected Problems on Exceptional Sets (1967) |
| [5] | Watson, A Treatise on the Theory of Bessel Functions (1984) |
| [6] | Falconer, The Geometry of Fractal Sets (1984) · Zbl 0587.28004 |
| [7] | DOI: 10.1017/S0305004100029236 · doi:10.1017/S0305004100029236 |
| [8] | Rogers, Hausdorff Measures (1970) |
| [9] | Falconer, Mathematika 32 pp 191– (1985) |
| [10] | Falconer, Mathematika 31 pp 25– (1984) |
| [11] | Steinhaus, Fund. Math 1 pp 93– (1920) |
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