# Successive minima and the basis of lattice

I am able to prove the following two propositions: Recall that the $$i$$-th successive minimum of $$L\in \mathcal L$$, denoted $$\lambda_i(L)$$ is the infimum of the radii of the balls containing $$i$$-linearly independent vectors in $$L$$. All norms are Euclidean 2-norm.

Let $$\Lambda$$ be a lattice $$\mathbb R^d$$. Assume that $$d\le 4$$, then there exist a basis $$\{v_1,\dots,v_d\}$$ of $$\Lambda$$ such that $$\|v_j\|=\lambda_j(\Lambda), ~\text{for}~ j=\{1,2,\dots,d\}.$$

and

Let $$\Lambda$$ be a lattice in $$\mathbb R^d$$. Then there exist a basis $$v_1,v_2,\dots,v_d$$ of $$\Lambda$$ such that $$\|v_1\|=\lambda_1(\Lambda),\|v_2\|_d \asymp_d \lambda_2(\Lambda),\dots,\|v_d\| \asymp_d \lambda_d(\Lambda).$$

But I am very puzzled if the follow combination of the above two is true:

Let $$\Lambda$$ be a lattice in $$\mathbb R^d$$. Then there exist a basis $$v_1,v_2,\dots,v_d$$ of $$\Lambda$$ such that $$\|v_i\|=\lambda_i(\Lambda) (1\le i \le 4),\|v_5\|_d \asymp_d \lambda_5(\Lambda),\dots,\|v_d\| \asymp_d \lambda_d(\Lambda).$$

Is the last statement correct? I don't see how it follows from the previous two statements directly. There is some magic with the dimension 4. The example $$\text{Span}\{e_1,e_2,\dots, e_d, \frac{1}{2}(e_1+\dots+e_d)\}$$ provides the obstruction for the reduction process for $$d\ge 5$$.

Jacques Martinet has a paper that proves a result like this "on average". For a lattice $$\Lambda$$, define $$H_b(\Lambda) = \min_{\{v_i\}_i\text{ a basis of }\Lambda}\frac{\prod_{i}^n \lVert v_i\rVert_2}{\det \Lambda}$$, $$M(\Lambda) = \frac{\prod_{i = 1}^n \lambda_i(\Lambda)}{\det \Lambda}$$, and $$Q_b(\Lambda) = \frac{H_b(\Lambda)}{M(\Lambda)}$$.

Then

Theorem 1.2 For $$n\geq 4$$, we have that $$Q_b(\Lambda) \leq \sqrt{5/4}^{n-4}$$

Theorem 1.3 For $$4\leq n\leq 8$$, we have that $$Q_b(\Lambda) \leq\sqrt{n}/2$$.

Note that Martinet's quantities are actually the square of what I quote, but his convention is to define things in terms of the squares of the relevant norms.

Anyway, I say this establishes your result "on average" as the two sequences can only differ "on average" by a constant multiplicative factor $$\sqrt{5/4}$$ on each term.

• Thanks for pointing out this interesting reference! Oct 23, 2022 at 15:48