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Let $B := [-1,1]^n$ be an $n$-dimensional box. Moreover, let $v_1,\ldots,v_n \in \mathbb{R}^n$ form a basis of $\mathbb{R}^n$, where the entries of the $v_i$ are explicitly irrational. We can assume that $\mathrm{det}(v_1,\ldots,v_n) = 1$.

Then, $$L := \left\{ \sum_{i=1}^n v_i k_i, \text{ where } (k_1,\ldots,k_n) \in \mathbb{Z}^n \right\} $$ is a lattice.

I am interested in enumerating (not just counting) all the points in $L \cap B$ for $n$ up to dimension 12. Is this possible at all? Which algorithms are suited for this problem?

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  • $\begingroup$ Take suitable rational approximations and use BKZ (or LLL), then generate points and test them until you are guaranteed to be outside the box. $\endgroup$
    – Steve Huntsman
    Commented Aug 27, 2015 at 0:45
  • $\begingroup$ The rational approximation should do the trick. Then you can trianguliarize your basis with Hermite Normal Form and backtrack the coordinate of the points. $\endgroup$
    – Campello
    Commented Dec 9, 2016 at 0:44

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