The Hadwiger–Nelson problem asks for the minimum number of colors required to color the plane such that no two points at distance 1 from each other have the same color.

We could build the following discrete variant on the infinite two dimensional **integer grid** (the nodes are points with integer coordinates).

[Discrete Hadwiger–Nelson problem variant]Given an integer $U$ (which is the equivalent of theunit distancein the Hadwiger–Nelson problem), and an integer tolerance distance $t < U$; we consider the infinite induced subgraph $G_{U,t}$ of the integer grid in which two integer points $(x_1,y_1), (x_2, y_2)$ are linked if their distance $d$ is in the interval $[U-t, U+t]$ ( $$|U - ((x_1 - x_2)^2 + (y_1 - y_2)^2)^{1/2}| <= t$$ What is the chromatic number of $G_{U,t}$ ?

Is this problem known?

If it is known, can you give some references?