Homogeneous iterative sampling in multidimensional space?

Question

Algorithms

There is a function of Nd parameters f(x[1], x[2], x[3],…, x[Nd]). Boundary values ​​are given for each of the parameters. I want to iteratively calculate the values ​​of this function so that the farther, the greater the density of the iteration points, with the density growing evenly for all arguments, in the entire space.
In one dimension, the following algorithm immediately comes to mind: we take a segment, divide by 2, each resulting segment by 2, and so on, reduce the sampling interval. Tell me how you can effectively implement a similar approach in a multidimensional space, and these calculations will go in parallel, which should not be difficult.
How to implement such an iterator, so that the further it grows, the denser the sampling goes? How to calculate what values ​​you need to take for the function parameters depending on the iteration number, so that you get such a homogeneous sampling with gradually increasing density?
You can use a random number generator, but then the more iterations, the greater the likelihood of local compactions / voids, and my task is to ensure the maximum minimum density, so that it would not be dense in one place, empty in another.

Answer 1

Like thought up, for a one-dimensional case:
{0/1, 1/1}; {1/2}; {1/4, 3/4}; {1/8, 3/8, 5/8, 7/8}; {1 / 16.3 / 16.5 / 16.7 / 16.9 / 16.11 / 16.13 / 16.15 / 16}; ...
That is, in the numerator we sequentially go through all the odd ones, in the denominator the deeper, the greater than a power of two. There is no need for any if and storage of "passed" values.

Answer 2

-