The Hyperboloid Solution

Every sufficiently large integer n can be written as x² + y² − z² with max(x², y², z²) ≤ n. This is Erdős Problem 1148 — not just the existence of the representation (which is easy) but the boundedness constraint on the components.

Chojecki resolves it by moving the problem to a hyperboloid.

The equation x² + y² − z² = n defines a one-sheeted hyperboloid in three dimensions. Points with integer coordinates on this surface are the solutions. The boundedness constraint selects a compact region of the surface. The question becomes: does this region always contain at least one integer point?

Duke’s theorem on equidistribution of CM points on modular curves provides the answer. Integer points on the hyperboloid are equidistributed with respect to the surface measure, so any region of sufficient area contains points. The bounded region around n has area proportional to n, which grows — eventually exceeding the threshold for guaranteed equidistribution.

The structural move: an arithmetic problem (write n as a sum of three squares with signs) becomes a geometric problem (find an integer point in a region of a hyperboloid), which becomes an equidistribution problem (are lattice points spread evenly on the surface?). Each translation tightens the analysis.

The bounded representation existed because the hyperboloid is curved just right — its curvature ensures that integer points can’t clump or avoid any particular region. The geometry forces the arithmetic to cooperate.