We show that the Hausdorff dimension of any slice of the graph of the Takagi function is bounded above by the Assouad dimension of the graph minus one, and that the bound is sharp. The result is deduced from a statement on more general self-affine sets, which is of independent interest. We also prove that Marstrand’s slicing theorem on the graph of the Takagi function extends to all slices if and only if the upper pointwise dimension of every projection of the length measure on the x-axis lifted to the graph is at least one.

中文翻译：

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Takagi 函数的切片

我们证明，Takagi 函数图的任何切片的 Hausdorff 维数都以图的 Assouad 维数减一为界，并且该界限是尖锐的。结果是从更一般的自仿射集的陈述中推导出来的，这是独立的。我们还证明了 Takagi 函数图上的 Marstrand 切片定理扩展到所有切片当且仅当长度测量的每个投影的上逐点维度X提升至图表的 - 轴至少为 1。