Consider the interval $[p_k,p_k\#]$. It contains $(\phi(p_k\#))-1=(\prod_{i=1}^k (p_i -1))-1$ numbers relatively prime to $p_k\#$ (NB the $-1$ at the end of each expression is strictly necessary because the number $1$ is identified by the totient but does not fall in the interval). For $p_m<p_k\#<p_{m+1}$, $m-k$ of those numbers are prime. Let's give the name $A_k$ to the set of numbers in that interval relatively prime to $p_k\#$. The density of primes will be greater toward the low end of the interval (average gap between primes $\frac{p_k}{\ln(p_k)}$ vs. $\frac{p_k\#}{\ln(p_k\#)}$), and the density of composites will be greater toward the high end of the interval. For example, the smallest composites in that set will be $p_{k+1}^2$, then $p_{k+1}p_{k+2}$, etc., whereas there will be many primes between $p_k$ and $p_{k+1}^2$.
Question 1: Is there a mathematical description of the distribution of all members of set $A_k$ across the interval? For example, if we divide the numbers up to $p_k\#$ into $p_k$ subintervals each of length $p_{k-1}\#$, would we expect that the members of $A_k$ would be roughly equally distributed into the subintervals?
Now consider the product $\prod_{i=1}^k (p_i -2)$. This product counts the number of numbers $q$ in the interval $[p_k,p_k\#]$ for which both $q$ and $q+2$ are relatively prime to $p_k\#$. Note that since only $q$ needs to be in the interval, the pair $p_k\#-1,p_k\#+1$ is included in the count. The set of numbers $q$ (call it $B_k$) is a subset of $A_k$.
Question 2: Is there a mathematical description of the distribution of members of set $B$ across the interval?
The motivating thought behind these questions is as follows: If the distribution of members of $B_k$ is relatively uniform across the interval, then at the low end of the interval where composite members of $A_k$ are sparse, it might be demonstrable that at least one member of $B_k$ must be the smaller prime in a pair of twin primes. If any such argument could be generalized to every (or almost all) $p_k$, the twin prime conjecture might be proved. I realize that since this line of thought has not already been used to prove the twin prime conjecture, it is unlikely to bear fruit, but I would like to learn about the issues and limitations pertaining to this line of thought.
A question with a similar theme asks about twin primes in the interval $[(p_k)^2,(p_{k+1})^2]$, but I don't see that the answer it received illuminates my present questions.