math.CO — Combinatorics

An Exponent of 1.04273 for the Unit Distance Problem

Contributed by Eric Naslund

Let $u(U)$ count the unordered pairs at distance one in a finite planar set $U$. We construct finite sets $U_j$ with $|U_j|\to\infty$ and $u(U_j)/|U_j|^{1.04273}\to\infty$. The largest exponent previously claimed, $1.0358$, is in the author's unpublished manuscript. The method is the number-field construction of OpenAI and Sawin: unit distances come from elements of relative norm one in quadratic extensions, and the fields come from an infinite pro-$2$ class field tower. The new ingredients are quadratic extensions of mixed signature, with an exact average over their norm-one units, and a tower over the real quadratic field $\Q(\sqrt{241})$, in which $2$, $3$ and $5$ split. Its Golod--Shafarevich function contains two copies of the local conditions at these primes but only one constant term, and the extra room lets the tower be ramified only above $2$, $3$ and $5$; the root discriminant of its fields is about $286$. The relative zeta value is bounded through the zeta function of the degree-$512$ field generated over $\Q(\sqrt{241})$ by the square roots of its $\{2,3,5\}$-units, which is the product of the Dedekind zeta function of $\Q(\sqrt{241})$ and $255$ quadratic Hecke $L$-functions. Finite facts and numerical inequalities are certified by exact computation and interval arithmetic. A significant portion of this work was verified in Lean, reducing the result with exponent $1.0427$ to an explicit zeta function inequality (Palomar registry, PALOMAR-2026-10-01-000018, version 1).