Algorithmic Information Bounds for Distances and Orthogonal Projections

We develop quantitative algorithmic information bounds for orthogonal projections and distances in the plane. Under mild independence conditions, the distance $|x-y|$ and a projection coordinate pₑ x each retain at least half the algorithmic information content of x in the sense of finite-precision Kolmogorov complexity, up to lower-order terms. Our bounds support conditioning on coarser approximations, enabling case analyses across precision scales. The proofs introduce a surrogate point selection step. Via the point-to-set principle we derive a new bound on the Hausdorff dimension of pinned distance sets, showing that every analytic set E² with H(E)≤ 1 satisfies \[x∈ E_H(Δ_x E)≥ 3/4_H(E).\] We also extend Bourgain's theorem on exceptional sets for orthogonal projections to all sets that admit optimal Hausdorff oracles.