I am looking for a proof, disproof or counter example of the following claim.
Let $C = \{k_1, k_2, \ldots, \}$ be a strictly increasing infinite sequence of positive integers which have a certain characteristic $C$. For example $C$ can be the set of all natural numbers, or all odd numbers or all even numbers or all primes or all numbers not divisible by 4 etc.
Let $N(x)$ be the numbers of positive integers $\le x$ that have this characteristic $C$ i.e. $N(x) = \sum_{k_i \le x} 1$. For example the number of integers less than $x$ have the characteristic of being even or prime is $x/2$ and $\pi(x)$ respectively.
Conjecture: If $(\frac{k_1}{k_n}, \frac{k_2}{k_n}, \ldots, \frac{k_{n-1}}{k_n},1)$ approach uniform distribution in $(0,1)$ as $n \to \infty$ then,
$$ \sum_{k_i \le n}{n\choose k_i} \sim \frac{2^n}{n}N(n). $$
The progress on the case of primes is present in this MSE post.
Edit: The additional constraint on uniform distribution has been added in light of @Paul SInclair's comment and @bof's counter example for non-uniformly distributed $k_i/k_n$