Unit-circle method for a distinct-distance lower bound
= Unit-circle method for a distinct-distance lower bound
{title2=$|\Delta(P)|\gtrsim |P|^{2/3}$}
For every positive distance, draw equal-radius circles centred at the points of $P$. Their <incidences between points and curves> count the ordered pairs at that distance. The <Szemerédi–Trotter theorem for unit circles> bounds each distance class by $O(|P|^{4/3})$. Summing over all classes accounts for $|P|(|P|-1)$ pairs, proving $|\Delta(P)|\gtrsim |P|^{2/3}$.