自仿射集在直线上的投影
Projections of self-affine sets onto lines
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中文总结 AI 辅助
研究自仿射集在直线上的投影，通过指数分离、邻近性和强不可约性假设，证明自仿射测度投影的Hausdorff维数结论，加强邻近性假设后自仿射集本身也有相同结论，在平面中仅线性部分强不可约性即可，还得出相关推论。
中文摘要 AI 辅助
我们证明了关于\(\mathbb{R}^d\)中自仿射测度和集合的全方向Marstrand - Mattila投影定理。在指数分离以及线性部分的邻近性和强不可约性假设下，自仿射测度在每条直线上的投影具有预期的Hausdorff维数。若将邻近性假设加强为强挤压，则自仿射集\(X\)本身在无任何分离假设时也有相同结论。在平面中，仅线性部分的强不可约性就足够且是精确的。作为推论，若\(X\)的上Minkowski维数至多为1，则其Minkowski维数存在且等于Hausdorff维数，部分回答了关于自仿射集Minkowski维数是否存在的民间问题。
英文摘要
We prove an all-directions Marstrand-Mattila projection theorem for self-affine measures and sets in $\mathbb{R}^d$. Under exponential separation, together with proximality and strong irreducibility assumptions on the linear parts, the projection of a self-affine measure onto every line has the expected Hausdorff dimension. If the proximality assumption is strengthened to strong pinching, then the same conclusion holds for the self-affine set $X$ itself, without any separation assumption. In the plane, strong irreducibility of the linear parts alone suffices, and this is sharp. As a corollary, if $X$ additionally has upper Minkowski dimension at most one, then its Minkowski dimension exists and equals its Hausdorff dimension, giving a partial affirmative answer to the folklore question of whether the Minkowski dimension exists for every self-affine set.
