Given a lattice $L = \bigoplus_{i=1}^{m} \mathbb{Z}v_i$ (the $v_i$ are linearly independent vectors in $\mathbb{R}^n$) and a number $c > 0$, can one quickly compute or find a good estimate on the number of lattice vectors $v$ with $|v| \leq c$ without actually enumerating these vectors? The basis $v_1,\ldots, v_m$ of the lattice can be assumed to be LLL reduced.

Also asked at: https://mathoverflow.net/questions/71052/estimating-the-number-of-short-vectors-in-a-lattice

  • 2
    $\begingroup$ Simultaneous cross-posting on different sites is discouraged, please wait a few days to see if you get an answer on one site before posting it on another one. I am closing the question for now, wait a few days and if you still don't get a satisfying answer on the other site flag the question and we will reopen this question. $\endgroup$
    – Kaveh
    Commented Jul 23, 2011 at 12:40