Bisector energy and few distinct distances
We introduce the bisector energy of an $n$-point set $P$ in $\mathbb{R}^2$, defined as the number of quadruples $(a,b,c,d)$ from $P$ such that $a$ and $b$ determine the same perpendicular bisector as $c$ and $d$. If no line or circle contains $M(n)$ points of $P$, then we prove that the bisector energy is $O(M(n)^{\frac{2}{5}}n^{\frac{12}{5}+ε} + M(n)n^2).$. We also prove the lower bound $Ω(M(n)n^2)$, which matches our upper bound when $M(n)$ is large. We use our upper bound on the bisector energy to obtain two rather different results: (i) If $P$ determines $O(n/\sqrt{\log n})$ distinct distances, then for any $0<α\le 1/4$, either there exists a line or circle that contains $n^α$ points of $P$, or there exist $Ω(n^{8/5-12α/5-ε})$ distinct lines that contain $Ω(\sqrt{\log n})$ points of $P$. This result provides new information on a conjecture of Erdős regarding the structure of point sets with few distinct distances. (ii) If no line or circle contains $M(n)$ points of $P$, then the number of distinct perpendicular bisectors determined by $P$ is $Ω(\min\{M(n)^{-2/5}n^{8/5-ε}, M(n)^{-1} n^2\})$. This appears to be the first higher-dimensional example in a framework for studying the expansion properties of polynomials and rational functions over $\mathbb{R}$, initiated by Elekes and Rónyai.