We show that if compact set \(E\subset \mathbb {R}^d\) has Hausdorff dimension larger than \(\frac{d}{2}+\frac{1}{4}\), where \(d\ge 4\) is an even integer, then the distance set of E has positive Lebesgue measure. This improves the previously best known result towards Falconer’s distance set conjecture in even dimensions.

中文翻译：

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均匀尺寸下Falconer距离设置问题的改进结果

我们表明，如果紧集\（E \ subset \ mathbb {R} ^ d \）的Hausdorff维数大于\（\ frac {d} {2} + \ frac {1} {4} \），其中\（ d \ ge 4 \）是一个偶数整数，则E的距离集具有正的Lebesgue测度。这改善了以前对Falconer距离集猜想在均匀尺寸上的已知结果。