An improved bound on the number of dot products determined by a finite point set in the plane

We are interested to bound from below the number of distinct dot products determined by a finite set of points $P$ in the Euclidean plane. In this paper, we build on the work of B. Hanson, O. Roche-Newton, and S. Senger, to obtain the improved lower bound $$\mathrm{\mid }\{p\cdot q:p,q\in P\}\mathrm{\mid }\gtrsim \mathrm{\mid }P{\mathrm{\mid }}^{\frac{2}{3}+\frac{7}{1425}}\mathrm{.}$$