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On Falconer’s distance set problem in the plane. (English) Zbl 1430.28001

Falconer’s distance problem is about the connection between the Hausdorff dimension of a set \(E\subset {\mathbb R}^d\), with \(d\geq 2\), and the size of the associated distance set \(\{|x-y|, x,y\in E\}\). When \(E\) is compact, the problem is to understand how large its Hausdorff dimension needs to be to ensure that the Lebesgue measure of the distance set is positive. In [Mathematika 32, 206–212 (1985; Zbl 0605.28005)], K. J. Falconer proved that a sufficent condition for the latter is that the Hausdorff dimension of \(E\) is \(>(d+1)/2\). He also gave examples of sets with Hausdorff dimension \(\leq d/2\) with zero Lebesgue measure. The Falconer Distance Conjecture, i.e., if the Hausdorff dimension of \(E\) is \(>d/2\), then the Lebesgue measure of the distance set is positive still stands. In the case \(d=2\), T. Wolff [Int. Math. Res. Not. 1999, No. 10, 547–567 (1999; Zbl 0930.42006)] proved that if \(E\) has Hausdorff dimension greater than \(4/3\), then its distance set has positive Lebesgue measure. In this paper, the authors improve the bound to \(5/4\).

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
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