2-distance list $(Δ+ 3)$-coloring of sparse graphs

A 2-distance list k-coloring of a graph is a proper coloring of the vertices where each vertex has a list of at least k available colors and vertices at distance at most 2 cannot share the same color. We prove the existence of a 2-distance list $(\Delta + 3)$-coloring for graphs with maximum average degree less than $\frac83$ and maximum degree $\Delta\geq 4$ as well as graphs with maximum average degree less than $\frac{14}5$ and maximum degree $\Delta\geq 6$.