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On Falconer’s distance set conjecture. (English) Zbl 1141.42007

Let \(E\) be a compact subset of \(\mathbb{R}^d\) and \(\Delta(E)=\{|x-y| : x,y \in E\}\). Falconer’s conjecture is that if the Hausdorff dimension of \(E\) is greater than \(d/2\), then the Lebesgue measure of \(\Delta(E)\) is positive. In this paper the author proves that this is true if \(d > 2\) and \(\dim(E) > d(d+2)/2(d+1)\).

MSC:

42B10 Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type
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