Turán type results for distance graphs in infinitesimal plane layer

In this paper we obtain the lower bound of number of edges in a unit distance graph $\Gamma$ in an infinitesimal plane layer $\mathbb R^2\times[0,\varepsilon]^d$ which compares number of edges $e(\Gamma)$, number of vertices $\nu(\Gamma)$ and independence number $\alpha(\Gamma)$. Our bound $e(\Gamma)\ge\frac{19\nu\Gamma)-50\alpha(\Gamma)}3$ is generalizing of previous bound for distance graphs in plane and a strong upgrade of Turán's bound when $\frac15\le\frac{\alpha(\Gamma)}{\nu(\Gamma)}\le\frac27$.