I'm trying to convert a decimal, $0 \leq decimal \leq 1$ to another $0\leq weighted \leq 1$.
Such that for $n$ equally spaced fractions, the weighted fractions $\forall1\leq i\leq n: weighted_{\frac{i}{n}}$ become exponentially sparse as they approach 1.
I'm not sure what you mean by "exponentially sparse". To divide the unit interval into $n$ pieces that thin out as you near $1$, you could consider division points $$ x_i = \left(\frac{i}{n}\right)^M \quad i = 1, \cdots, n-1 $$ for some large $M$.
Another possibility is to divide the interval into subintervals such that each is $r$ times as large as the preceding one. Then if the first division point is at $x$ you have $$ x + rx + r^2x + \cdots + r^{n-1}x = 1 $$ so $$ x = \frac{r-1}{r^n - 1} . $$ Then choose $r$ to find $x$.
For example, if $r = 2$ and $n=3$ the division points are at $1/7$ and $3/7$ giving intervals of length $1/7, 2/7, 4/7$.