Erdős–Ulam problem

Three points at positions zero, one and two on a line have distances one, one and two. Can a set with fraction-valued distances between every pair also reach every tiny neighborhood throughout the plane? A whole line still leaves neighborhoods elsewhere empty.[1]

Can one set reach every small neighborhood of the plane while every distance between two of its points is a fraction? This is stronger than being infinite on a line. For example, $\{(q,0):q\in\mathbb Q\}$ has rational pairwise distances but misses a small disk around $(0,1)$.

Coordinates alone do not secure the distance rule: $(1,0)$ and $(0,1)$ are each distance one from the origin, yet their mutual distance is $\sqrt2$, which is irrational. A dense set must put a point in every open disk, however small, and satisfy the rule for every pair of its own points.[1] [2]

No construction or impossibility proof settles both requirements together.[2]

Solymosi and de Zeeuw prove restrictions on infinite rational-distance sets lying on algebraic curves, with lines and circles as special cases. The 2025 paper treats rational distances to fixed reference points; its conditional discussion does not prove a negative answer to the unrestricted pairwise problem.[1] [2]

Fractions impose exact arithmetic rules, while filling the plane permits geometric freedom. The problem tests how strongly those arithmetic rules restrict a cloud of points.[1]

Squared distance is an equation in coordinates, but rational distance additionally requires its square root to be rational. Requiring this for every pair ties many equations together. The problem connects how arithmetic equations restrict point sets with how densely those sets can occupy ordinary space.[1] [2]