Let $n$ be positive integer, $k$,$B$ fixed positive integers.

Let $f_i(x_1,x_2...x_n)$ be a system of $n-k$ linearly independent linear
equations over the integers.

Let $S(f_i,k,B)$ be the set of solutions of $f_i=0$.

>Q1 How large can $S(f_i,k,B)$ be subject to the constraints $|x_i| \le B$?
Can it be unbounded for some $k,B$?

Let $T_1,T_2...T_n$ be sets of integers of size $B$.

>Q2 How large the solutions can be subject to the constraints $x_i \in T_i$?

We suspect for $B=2$ the set of solutions is small.


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Some experiments with selecting the points and finding the equations failed, since it returned linearly dependent $f_i$.