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I am looking for a certain notion of sparseness of lattices.

I want to find a vector in $\mathbb{Z}^N$ that the minimal possible inner product with all the vectors of a given lattice. Or at least, I would like to know what this number is.

In other words, for a positive-definite lattice $\Gamma$ or rank $N$, what is $$ \min_{\vec{v} \in \mathbb{Z}^N} \max_{\vec{u} \in \Gamma} \frac{\vec{u} \cdot \vec{v}}{u v} $$

If $\Gamma$ is a hypercubic lattice, then I can always choose $\vec{u}$ to be $\vec{v}$, which means this quantity is just 1. Can I choose $\Gamma$ such that it is provably less than 1?

What is the value of this quantity for the smallest unimodular even lattice ($E_8$)?

I would expect that unimodular lattices with no roots (vectors of norm 1 or 2) are sparse in the above sense; the intuition being that they miss many of the short vectors in $\mathbb{Z}^N$.

Edit: normalization

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    $\begingroup$ I am having trouble parsing the definition of the quantity in question. Why wouldn't the maximum over $u$ be infinite? $\endgroup$
    – Hugh Thomas
    Commented Apr 7, 2014 at 2:39
  • $\begingroup$ Also, how is $\mathbb{Z}^N$ related to $\Gamma$? Is $\Gamma = \mathbb{Z}^N$? $\endgroup$
    – Simon Rose
    Commented Apr 7, 2014 at 2:47
  • $\begingroup$ Indeed, I forgot to divide by the length of $u$ itself. Fixed now. The only relationship between $\Gamma$ and $\mathbb{Z}^N$ is that they have the same rank. The point of my question is to understand what the above quantity can be depending on $\Gamma$. $\endgroup$
    – Eugeniu Plamadeala
    Commented Apr 7, 2014 at 3:59
  • $\begingroup$ For any choice of $\vec{v}$, there will be $\vec{u} \in \Gamma$ with arbitrarily small angular separation from the line $\mathbb{R}\vec{v}$. This means your quantity is always 1. Do you want to restrict $\vec{v}$ to be in some ball? $\endgroup$
    – S. Carnahan ♦
    Commented Apr 7, 2014 at 8:31
  • $\begingroup$ Thank you Scott. I see now that inner product can be made arbitrarily close to 1. $\endgroup$
    – Eugeniu Plamadeala
    Commented Apr 7, 2014 at 18:37

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