A power saving over the Spencer-Szemerédi-Trotter n^(4/3) bound for planar unit distances (Erdős Problem 90)

Theorem 1.1: there are absolute constants $0<C<\infty$ and $1\le\beta<4/3$ such that every set of $n$ points in the Euclidean plane has at most $Cn^\beta$ unordered pairs at distance $1$, for every $n$. This is the first improvement of the 1984 exponent $4/3$. It leaves a gap to the lower bound $n^{1.0141}$ (Sawin) from the 2026 disproof of Erdős's $n^{1+o(1)}$ conjecture, and $\delta=4/3-\beta$ is not made explicit. Not addressed: distinct distances, higher dimensions, or other norms (for Valtr's metric $4/3$ is sharp).

Theorem 1.1: there are absolute constants $0<C<\infty$ and $1\le\beta<4/3$ such that every set of $n$ points in the Euclidean plane has at most $Cn^\beta$ unordered pairs at distance $1$, for every $n$. This is the first improvement of the 1984 exponent $4/3$. It leaves a gap to the lower bound $n^{1.0141}$ (Sawin) from the 2026 disproof of Erdős's $n^{1+o(1)}$ conjecture, and $\delta=4/3-\beta$ is not made explicit. Not addressed: distinct distances, higher dimensions, or other norms (for Valtr's metric $4/3$ is sharp).