We investigate the relationship between algorithmic fractal dimensions and the classical local fractal dimensions of outer measures in Euclidean spaces. We introduce global and local optimality conditions for lower semicomputable outer measures. We prove that globally optimal outer measures exist. Our main theorem states that the classical local fractal dimensions of any locally optimal outer measure coincide exactly with the algorithmic fractal dimensions. Our proof uses an especially convenient locally optimal outer measure $\boldsymbol{\kappa}$ defined in terms of Kolmogorov complexity. We discuss implications for point-to-set principles.

中文翻译：

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算法优化的外部度量

我们研究了欧几里德空间中算法分形维数与外部度量的经典局部分形维数之间的关系。我们为较低的半可计算外部度量引入了全局和局部最优条件。我们证明了全局最优的外部度量存在。我们的主要定理指出，任何局部最优外部度量的经典局部分形维数与算法分形维数完全一致。我们的证明使用了一个特别方便的局部最优外部度量 $\boldsymbol{\kappa}$，定义为 Kolmogorov 复杂度。我们讨论了点对集原则的含义。