An inequality is derived for the average energies of pinned distance measures. This refines Mattila’s theorem on distance sets (Mattila in Mathematika 34:207–228, 1987) to pinned distance sets, and gives an analogue of Liu’s theorem (Liu in Geom Funct Anal 29:283–294, 2019) for pinned distance sets of dimension smaller than 1.

中文翻译：

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通过球面平均值的低维固定距离集

固定距离量度的平均能量不等式导出。这将Mattila的距离集定理（Mattila in Mathematika 34：207–228，1987）细化为固定距离集，并给出了Liu定理的类似物（Liu in Geom Funct Anal 29：283–294，2019）类似的固定距离集。尺寸小于1。