Let $\sum_{i=1}^k a_ix_i = N$ be the equation with $a_i \in [2^t,2^{t+1}]$ being distinct primes. Let region $R_i = (0,a_i)\cap \Bbb Z$. Then if we seek solutions unique $x_i\in R_i$, then in general it is not possible.

However, is it possible to divide each interval $R_i$ into $f(k,t)$ non-overlapping intervals for some polynomial $f$ such that the total number of integer points in $S_i$, the $\underline{\mbox{union of a subset}}$ of the non-overlapping intervals, satisfies $|S_i|>{2^{t-\delta(t,k)}}$ for some function $\delta(t,k)$ whose range on positive $t,k$ is bounded below by $0$ such that if we seek solutions $\sum_{i=1}^k a_ix_i = N$ with $x_i\in S_i$ then solution is unique? What is the function $\delta(t,k)$ and how to find the solution $x_i\in S_i$?