Distinct-distance set ($\Delta (P)=\{|p-q|:p,q\in P\}$)

For a finite planar point set $P$, its distinct-distance set is $\Delta (P)=\{|p-q|:p,q\in P\}$. It includes zero for nonempty $P$. Its cardinality measures how many different distances the configuration determines. Ordered point pairs can be grouped according to their distance, which relates the problem to incidences between points and curves.

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 $|\Delta (P)|\gtrsim |P{|}^{2/3}$.