In this paper, we use algorithmic tools, effective dimension and Kolmogorov complexity, to study the fractal dimension of distance sets. We show that, for any analytic set $E\subseteq\R^2$ of Hausdorff dimension strictly greater than one, the \textit{pinned distance set} of $E$, $\Delta_x E$, has Hausdorff dimension of at least $\frac{3}{4}$, for all points $x$ outside a set of Hausdorff dimension at most one. This improves the best known bounds when the dimension of $E$ is close to one.

翻译：在本文中,我们使用算法工具、有效维度和Kolmogorov复杂度来研究距离设置的分形维度。我们显示,对于任何分析集,Hausdorf维度的$E\subseteq\R ⁇ 2$严格大于1,当美元维度接近于1时,对于Hausdorf维度以外的所有点,Hausdorf维度至少为$\frac{3 ⁇ 4}$x$。这改善了已知的最佳界限。