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 $R_i = (0,a)\cap \Bbb Z$. 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 disjoint regions $S_i$ and $S_i^c$ such that the number of integer points in $S_i$ 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 unique and how to find the solution?