  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.
