In my book about Lattice basis reduction I found the following:

Gaussian Heuristic: for $\displaystyle r$ big enough, there are approximately $\displaystyle $v_m r^m /vol(L)$$ lattice points in $\displaystyle $L$$ of norm $\displaystyle $\le r$$, where $\displaystyle $v_m$$ denotes the volume of the m-dimensional unit ball.

Corollary: for a lattice $\displaystyle $L$$ of rank $\displaystyle $m$$ the shortest nonzero lattice vector has norm at most approximately $\displaystyle $vol(L)^{1/m}$$.

Then they give this example:

Given $\displaystyle $\alpha, \beta, \gamma \in \mathbb{R}$$, and $\displaystyle $X$$ > $\displaystyle $0$$ with $\displaystyle $X$$ large. Small values for the linear form $\displaystyle $f(x,y,z)=\alpha x + \beta y + \gamma z$$ with $\displaystyle $x,y,z \in \mathbb{Z}$$ corresponds to lattices

$\displaystyle \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0\\ C\alpha & C\beta & C\gamma \end{pmatrix}$

How large should $\displaystyle $C$$ be taken to compute $\displaystyle $x,y,z \in \mathbb{Z}$$ with $\displaystyle $max(\mid x \mid,\mid y \mid,\mid z \mid) \le X$$ and $\displaystyle $\mid \alpha x + \beta y + \gamma z \mid$$ minimal?

Their answer is:

The Gaussian heuristic with $\displaystyle $C=X^3$$ expects solutions of size approximately $\displaystyle $X$$, with $\displaystyle $\mid \alpha x + \beta y + \gamma z \mid $$ approximately $\displaystyle $X^{-2}$$.

How do they come up with $\displaystyle $C=X^3$$ ?

Using the above Corallary I have that the shortest nonzero lattice vector has norm at most approximately $\displaystyle $vol(L)^{1/m} = \mid determinant(L) \mid ^{1/3}=(C\gamma)^{1/3}$ $

Then since $\displaystyle $max(\mid x \mid,\mid y \mid,\mid z \mid) \le X$$ we have that the norm of $\displaystyle $(\alpha x, \beta y, \gamma z) \le \sqrt(\alpha^2 + \beta^2 + \gamma^2) X$.

So $\displaystyle $(C\gamma)^{1/3} \le \sqrt(\alpha^2 + \beta^2 + \gamma^2) X$ $and thus $\displaystyle $C$$ is approximately $\displaystyle $X^3$ $?

Am I doing this right? I hope someone can explain where the$\displaystyle $C$$ is coming from.

Thank you!