Solutions to $\gamma \equiv \sum_{i=1}^m \xi_i\cdot x_i\bmod p$ with $|x_i| < \ell$
Are there any clear conditions on $p,\ell$ and $m$ under which the equation $\gamma \equiv \sum_{i=1}^m \xi_i\cdot x_i\bmod p$ has at most one solution with $|x_i|&lt;\ell$, where $\gamma$ and the $\xi_i$ are uniformly random?


For any $\ell \ge 2$, there will be values of $\gamma$ and $\xi_i$ for which there are multiple solutions; one such set of values would be $\gamma = 0$ and $\xi_1 = 0$.  Hence, any condition that forbids multiple solutions would necessarily have $\ell &lt; 2$ (which isn't very interesting)

Perhaps you mean to ask "yields more than one solution with probability $&lt; \epsilon$"; that is a rather harder question, but might give you a more useful answer.