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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.

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

This is related to the question Can we find curves with many rational points using linear algebra?

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  • $\begingroup$ Probably Siegel lemma will be relevant in this situation. $\endgroup$
    – Alexey Ustinov
    Commented Nov 11 at 11:56
  • $\begingroup$ @AlexeyUstinov Thanks. Getting many small solutions might be of theoretical interest. The lemma appears to give worst case upper bound, while I am asking for small solutions. $\endgroup$
    – joro
    Commented Nov 11 at 12:52

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