A power saving over Dey's bound for planar halving lines

Theorem 1.1: there are absolute $\varepsilon>0$, $C$, $n_0$ with $h(P)\le Cn^{4/3-\varepsilon}$ for every even $n\ge n_0$ and every $n$-point planar set with no three collinear. Theorem 1.2 bounds the switches at every rank uniformly by $CN^{4/3-\varepsilon}$, and Corollary 1.3 gives $O(n(k+1)^{1/3-\varepsilon_0})$ $k$-sets for $1\le k\le n/2$ (also for levels in line arrangements). The constants and exponent are ineffective; no numerical value of $\varepsilon$ is given. It does not determine the order of $h(n)$, which could still lie anywhere between $n e^{\Omega(\sqrt{\log n})}$ and $n^{4/3-\varepsilon}$.

Theorem 1.1: there are absolute $\varepsilon>0$, $C$, $n_0$ with $h(P)\le Cn^{4/3-\varepsilon}$ for every even $n\ge n_0$ and every $n$-point planar set with no three collinear. Theorem 1.2 bounds the switches at every rank uniformly by $CN^{4/3-\varepsilon}$, and Corollary 1.3 gives $O(n(k+1)^{1/3-\varepsilon_0})$ $k$-sets for $1\le k\le n/2$ (also for levels in line arrangements). The constants and exponent are ineffective; no numerical value of $\varepsilon$ is given. It does not determine the order of $h(n)$, which could still lie anywhere between $n e^{\Omega(\sqrt{\log n})}$ and $n^{4/3-\varepsilon}$.