We establish a first general partial regularity theorem for area minimizing currents \({\mathrm{mod}}(p)\), for every p, in any dimension and codimension. More precisely, we prove that the Hausdorff dimension of the interior singular set of an m-dimensional area minimizing current \({\mathrm{mod}}(p)\) cannot be larger than \(m-1\). Additionally, we show that, when p is odd, the interior singular set is \((m-1)\)-rectifiable with locally finite \((m-1)\)-dimensional measure.

中文翻译：

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最小电流的面积规律mod p

我们建立了第一个通用的部分规则性定理，用于在任何维度和维数下，针对每个p最小化电流\（{\ mathrm {mod}}（p）\）。更确切地说，我们证明了最小化电流\（{\ mathrm {mod}}（p）\）的m维区域的内部奇异集的Hausdorff维不能大于\（m-1 \）。此外，我们证明了，当p为奇数时，内部奇异集可通过局部有限\（（m-1）\）-维测度进行\（（m-1）\）可纠正。