The weak pinned Erdős distance conjecture: some point determines n^(1-o(1)) distinct distances (Erdős Problem 604, first part)

Theorem 1.1: for every fixed $s>0$, $F_n(s)\to0$, where $F_n(s)$ is the largest fraction, over $n$-point planar sets, of ordered distinct pairs $(x,y)$ such that at least $n^s$ points of the set lie at distance $\|y-x\|$ from $x$. Corollary 1.2: for every fixed $\varepsilon>0$ the fraction of pins with fewer than $n^{1-\varepsilon}$ distinct distances tends to $0$ uniformly, so every large set has such a pin, and in fact all but $o(n)$ points are. No explicit rate is given. Not shown: the stronger form $\gg n/\sqrt{\log n}$ in Erdős's question, the averaged conjecture $\sum_x|D_x|\gg n^2/\sqrt{\log n}$, or any quantitative form of the $n^{o(1)}$ loss.

Theorem 1.1: for every fixed $s>0$, $F_n(s)\to0$, where $F_n(s)$ is the largest fraction, over $n$-point planar sets, of ordered distinct pairs $(x,y)$ such that at least $n^s$ points of the set lie at distance $\|y-x\|$ from $x$. Corollary 1.2: for every fixed $\varepsilon>0$ the fraction of pins with fewer than $n^{1-\varepsilon}$ distinct distances tends to $0$ uniformly, so every large set has such a pin, and in fact all but $o(n)$ points are. No explicit rate is given. Not shown: the stronger form $\gg n/\sqrt{\log n}$ in Erdős's question, the averaged conjecture $\sum_x|D_x|\gg n^2/\sqrt{\log n}$, or any quantitative form of the $n^{o(1)}$ loss.