For a certain problem, I need to make distance dependent statistics, but with the constraint that the number of sampling points, $N$, should be kept as small as possible. To be more specific I need to get a set of points in $\mathbb T^d$, ($\mathbb T=\mathbb R/\mathbb Z$ is the one dimensional torus, and $\mathbb T^d$ the $d$ dimensional torus) $$S=\{x_1,\,x_2,\dots,\,x_N\}\subset\mathbb T^d$$ such that the distances $$D=\{\|x-y\|, \text{such that}\; x,y\in S, x\neq y\}$$ are as evenly distributed as possible on $[0,\frac12\sqrt d]$.
I have started to create the set $S$ by picking random points uniformly. Unfortunately, this is not satisfying because the distribution of distances $r$ is quite uneven (see the figure displaying the distribution of distances between two random points in $\mathbb T^3$). As a result, short and large distances are almost not sampled. I have then tried to use random walks to cure the short distance problem, but it is not very efficient and it is still bad for large distances.
Is there a way to chose the elements of $S$ more efficiently ?