Assume we have a sequence $a_1,\dots,a_k\in\Bbb N$ with each $a_i\approx n$ where $n$ is some integer.

Suppose there exists an $\eta\in(0,1)$ such that for every $d_1,\dots,d_k\in\Bbb Z$ with $$a_1d_1+\dots+a_kd_k=0$$ there is a $d_i$ with $|d_i|>n^\eta$ then is it true that for every prime $p\approx n^{\ell'+\eta}$ where $1\leq\ell'$ holds there is an $m\in\Bbb N$ with $n^{\eta+\ell'-1}<m<p-n^{\eta+\ell'-1}$ such that each of $$ma_i\bmod p$$ lie in $[0,n^{\ell'}]$?

Is there a way to get such a result from discrepancy theory and Erdos-Turan-Koksma result?