Secret sharing over reals - constructing a (k,n) threshold scheme
This $(k,n)$ scheme works, but isn't very interesting.

Effectively, it is:


For each set of $k$ participants out of $n$, construct a $(k,k)$ threshold scheme, and distribute those shares to the participants in the set.


For example, in a $(2, 3)$ scheme, if $z$ is the secret, we'd generate $\binom{3}{2} = 3$ indepedent $(2,2)$ threshold schemes $(r_1, z-r_1 \bmod 1), (r_2, z-r_2 \bmod 1), (r_3, z-r_3 \bmod 1)$, and distribute to the three share holders the shares:

$$(r_1, z-r_2 \bmod 1)$$
$$(r_2, z-r_3 \bmod 1)$$
$$(r_3, z-r_1 \bmod 1)$$

It works, as:


For any set of $k$ share holders, they can reconstruct the secret, as there is a $(k,k)$ threshold scheme with those share holders with all the shares.  In the above example, the first two share holders jointly know both $z-r_2 \bmod 1$ and $r_2$, allowing them to reconstruct $z$.
For any set of $k-1$ share holders, they learn nothing of the secret; for any of the $(k,k)$ threshold schemes, there will always be at least 1 missing share, and so they can learn nothing.


It's quite straight-forward; the biggest issue is that this requires distributing $\binom{n-1}{k-1}$ independent values to each share holder; this is rather large if you're trying to implement a $(500,1000)$ access structure.