Bisector Energy and Few Distinct Distances

• DOI: 10.1007/s00454-016-9783-5
• Publication Date: 2016-06-07T23:36:22Z
• Authors: Ben Lund; Adam Sheffer; Frank de Zeeuw
• Keywords: Isosceles triangle

18 pages, 2 figures<br/>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 $O(M(n)^{\frac{2}{5}}n^{\frac{12}{5}+��} + M(n)n^2).$. We also prove the lower bound $��(M(n)n^2)$, 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<br/>