Fix some nonzero integers $a_1, \dots, a_k$, with $k \geq 2$, and some real numbers $c_1, \dots, c_k \in (0,1)$.
For every positive integer $m$, let $N(m)$ be the number of $k$-tuple of integers $(x_1, \dots, x_k)$ such that $$a_1 x_1 + a_2 x_2 + \cdots + a_k x_k \equiv 0 \pmod m $$ and $0 \leq x \leq c_i m$ for $i=1,\dots,k$.
I am interested in the asymptotic behavior of $N(m)$ as $m \to \infty$.
My guess is that, for $m$ relatively prime with $a_k$, we have that $N(m)$ has order about $c_1 \cdots c_k m^{k-1}$. Since for $i=1,\dots,k-1$ we have about $c_i m$ choices for each $x_i$, while $x_k \equiv -a_k^{-1}(a_1 x_1 + \cdots + a_{k-1}x_{k-1}) \pmod m$ has "probability" about $c_k$ of being in $[0, c_k m]$. If $\gcd(m, a_k) > 1$ one can get a similar heuristic after reducing by the common divisor.
Are there some asymptotic formulas or good upper/lower bounds for $N(m)$ as $m \to \infty$ ?
Thanks for your help
