On k-degenerate Distance Graphs in Finite Fields and a Class of 2-degenerate Distance Graphs in the Integer Lattice

We give conditions on dimension and density to guarantee isometric copies of k-degenerate distance graphs in the model setting of subsets of finite fields. These arguments are then translated and used in conjunction with the circle method to guarantee isometric copies of arbitrarily large dilates of a class of 2-degenerate distance graphs in subsets of the integer lattice.