An improved constant factor for the unit distance problem

We prove that the number of unit distances among $n$ planar points is at most $1.94\cdot n^{4\mathrm{/}3}$, improving on the previous best bound of $8n^{4\mathrm{/}3}$. We also give better upper and lower bounds for several small values of $n$. We also prove some variants of the crossing lemma and improve some constant factors.