Generating a grid that scales nicely

Let two positive values $$x > y$$ be given along with an integer $$h$$. Consider the interval $$[0, x]$$. I want to split this up into $$h$$ subintervals each of equal length. This is of course always possible and there's a unique and obvious way of doing it.

But I have two extra requirements. Write the subintervals like $$[x_i, x_{i+1}]$$. The two requirements are then

1. The value $$y$$ must lie on a border of one of these subintervals, i.e. there must be an $$i$$ for which $$x_i = y$$.
2. If we instead had to create twice as many subintervals (so, $$2h$$), then every border $$x_i$$ from before is again a border now. So, for example, if in the $$h$$-subinterval construction, we had $$x_i$$ as a border, then $$x_i$$ is again a border in the $$2h$$-construction.

In order to accomplish this, we have one degree of freedom: $$x$$. What that means if that if it works for the current value of $$x$$, then that's what we want, but if it doesn't work, then we are allowed to change $$x$$ to some other value $$x' \approx x$$.