How can we best find a numerical solution to a system of $m\ge2$ linear diophantine modular inequalities $$\big((a^j x+b_j)\bmod n\big)<c\;\text{ for }1\le j\le m$$
where $x$ is the only unknown, $\;1\le c<n$, $\;\gcd(a,n)=1$, $\;a^j\not\equiv1\pmod n$ ?
Obviously, each inequality $j$ has exactly $c$ solutions in $x$, trivialy computed as $$x=(a^{-j}d)\bmod n\text{ for }n-b_j\le d<c+n-b_j$$ and therefore for constant $m$ there is an obvious method of cost $\mathcal O\big(c(\log n)^2\big)$, where we sieve the $c$ solutions for the first inequality.
I'm thus interested only in much better methods, e.g. of cost polynomial in $\log n$ regardless of $c$, for rather arbitrary $b_j$ except that they are such that a solution exists, and $m\log c\over(m-1)\log n$ slightly below $1$ (making it plausible that there's a single solution).
I'm ready to be convinced that the problem reduces to what's considered by Alan M. Frieze, Johan Hastad, Ravi Kannan, Jeffrey C. Lagarias, Adi Shamir: Reconstructing truncated integer variables satisfying linear congruences, in SIAM Journal on Computing, Volume 17 Issue 2, 1988, also freely available from the first author's website; but I at least need a hint on how to proceed, and would much prefer a self-contained answer (except for the part using the LLL, if that's required).