Erdős distinct distances problem

Four corners of a unit square give six pairs but only two distances: 1 and √2. In a 3×3 grid, 36 pairs give just five distances. Erdős asks how far such repetition can go when the number of points grows.[1] [2]

The four corners of a unit square have six point pairs but only the lengths $1$ and $\sqrt2$. Can an arbitrary large planar set reuse distances even more efficiently than a square grid? The open issue is the sharp logarithmic factor in the minimum, not how to count pairs for one distance.

For finite P⊂ℝ² with |P|=n≥2, let D(P)=|{|p−q|:p,q∈P,p≠q}|. The target is D(P)=Ω(n/√log n), uniformly in P. The logarithm base only changes the constant. Square-grid constructions give the matching upper order; constants and exact finite minima are outside this page’s target.[1] [2]

The target lower bound is a constant times n/√log n for every planar arrangement. Square grids reach that order. The proved general lower bound is n/log n, so a logarithmic-factor gap remains even though the main power of n is settled.[1] [2]

The universal theorem gives Ω(n/log n). Removing the remaining √log n loss is open. The 2026 Pfaffian-curve result gives stronger bounds under fixed-complexity and no-line/no-circle-arc restrictions, so it does not settle the universal minimum. Continuous Hausdorff-dimension distance questions and unit-distance multiplicities are different formulations.[1] [2]

Repeating a length means that different point pairs can be matched by a distance-preserving motion. This turns a counting puzzle into a question about how those motions fit together. It connects repeated distances to the broader study of intersections among geometric objects, which supplies tools for controlling coincidences without restricting points to a grid.[1] [2]

Repeated distances reveal more than a short list of lengths: they measure how much of a point set can line up with itself after a translation or rotation. If a motion makes k points overlap, each ordered pair among those points gives an equal-length comparison. Thus local coincidences in lengths can be studied as partial symmetries of the whole arrangement. This connects the question to incidence problems about lines meeting in three-dimensional space, where concentration on planes or ruled surfaces can produce many intersections. It relates repeated measurements to the structure of an entire configuration, rather than treating every equal pair as an isolated coincidence. The original points need not lie on a chosen curve.[1] [2]