For a threshold point over n points we can assign "weights" or "number of votes per key":
(T₁,w₁), (T₂,w₂), ...,(Tₙ,wₙ).
Instead of using each Tᵢ only once, we also use the keys Tᵢ+H(Tᵢ|1)G, Tᵢ+H(Tᵢ|2)G, ..., Tᵢ+H(Tᵢ|wᵢ)G during the creation of the threshold point. So, if Alice learns Tᵢ she learns the key for all of its votes.
Let's define the total number of votes as N = sum{wᵢ}. Now, we can choose any threshold t < N by enumerating all sums of subsets of size t. We can apply the OR operator to these sums to condense the threshold condition into a single point.
This defines a t-of-N weighted threshold point.