Problem 3

We build a set of points in the following way. First we draw an infinitely long line starting at $(0,0)$ at thirty degrees to the horizontal. Then we draw a strip of width 1 around this line consisting of all points whose distance from the line is less than or equal to a half, this strip is the area between the two dotted lines below. Finally, for every point $(n,m)$ inside the strip, where $n$ and $m$ are whole numbers, we draw a red dot at the closest point on the line to $(n,m)$. The first part of the set is drawn below.

The set of dots is called a cut and project set and is well studied in both maths and physics.

One feature of the set is that the distance from a dot to its nearest neighbour will always be $1/2$, $\frac{\sqrt{3}}{2}$ or $\frac{1+\sqrt{3}}{2}$.

If we do the same construction with a much wider strip, say width 2, then there are points at distance less than $1/2$ from each other. What is the smallest width w such that the corresponding set has pairs of points at distance $< 1/10$ from each other? Give your answer to three decimal places.