The range of each of successive minima for all unimodular lattices

Question

Let $\mathcal L$ be the space of unimodular (covolume one) lattices in $\mathbb R^d$. 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 $L$.

For example, the first successive minimum is the shortest nonzero vector in $L$ and the second successive minimum is the shortest vector in $L$ that is linearly independent of the first one.

I wonder if there are estimates/exact values for the following quantities for $1\le i \le d$:

\begin{gather*} A_i := \inf \{\lambda_i(L):L\in \mathcal L\}, \\ B_i := \sup \{\lambda_i(L):L\in \mathcal L\}. \end{gather*}

I know some of them from Minkowski's second convex body theorem, namely $B_1 < \infty$ and $A_d >0$. In addition, $A_r=0, r<d$ via the following example $ \begin{bmatrix} e^{-\frac{1}{d}t}I_{d-1} & 0 \\ 0 & e^{\frac{(d-1)t}{d}} \end{bmatrix} \mathbb Z^{d}$ as $t\to \infty$.

For these $2d$ quantities, I don't have to know their specific values, but I would like to know if they are finite/infinite/zero/nonzero. Are there ways (powerful theorems) to see they values? Also I haven't found this discussed in any books in geometry of numbers or Diophantine approximations.

Answer 1

While I do not have an answer, there are many pointers one can give that might be useful, too many for comments.

First, one can say better than $B_1 < \infty$. It is the Hermite constant, and one can get a tight (suitably random lattices come within a constant factor of it) upper bound of $\lambda_1(L) \leq O(\sqrt{n})$. One can pin down a precise constant upper bound (pair the link I've given with Stirling's inequality), but I do not feel like copying this.

Second, the "higher-dimensional" generalization of the hermite constant is not on the individual $\lambda_i(L)$, as you seem to hope, but instead on the quantity $\prod_{i = 1}^r \lambda_i(L)$. The upper bound of this over all covolume one lattices is known as Rankin's constant $\gamma_{n,r}$. Certain books discuss it, see for example Perfect Lattices in Euclidean Space section 2.8. There are some inequalities known, but I do not believe they are thought to be tight, like in the Hermite case. Note that there is a version of Minkowski's second theorem that actually gives a bound on Rankin's constant directly. It can be found in Martinet's book as cor. 2.6.9.

Third, you can still say something about your $B_i$. Well-rounded lattices are lattices for which $\lambda_1(L) = \lambda_n(L)$. Therefore, you can get lower bounds on $B_i$ by looking at the analogue of $B_i$ for WR lattices, which is simply the analogue of Hermite's constant for WR lattices. Briefly looking I can't find any sources on this, but you can at least get the bound $0<\mathsf{vol}(B_n(1/2))^{1/n} \leq B_i$, as the lattice defines a packing of balls of radius $1/2$ into $\mathbb{R}^n$. Note that $\mathsf{vol}(B_n(1/2))^{1/n} = o(1)$, so this is not even a constant lower bound. It is plausible/likely there is some well-known family of well-rounded lattices that, when normalized to have $\det L= 1$, give a constant lower-bound though. I'll leave you to investigate this though.

Finally, there are some other bounds relating the various $\lambda_i$'s. The most obvious thing to mention are so-called "transference results". These bound $1 \leq \lambda_i(L)\lambda_{n-i+1}(L^*)\leq O(n)$ for all $i$. The most famous one is due to Banaszczy in the 90's, due to applications in cryptography there have been some more recent developments (mostly tightening constants) that I cite as they are easier for me to find, see here.

There have also been some mild generalizations of successive minima known as "slopes" that are always close to the successive minima, but obey a nicer form of transference. See for example this survey for pointers.

Answer 2

For any numbers $\lambda_1, \dots, \lambda_d$ with $\lambda_1 \leq \lambda_2 \leq \dots \leq \lambda_d$ and $\lambda_1\lambda_2 \dots \lambda_d=1$, there exists a unimodular lattice with successive minima $\lambda_1,\dots, \lambda_d$, since we can just take the lattice of vectors whose $i$'th entry is an integer multiple of $\lambda_i$.

From this it is clear we can take $\lambda_i$ arbitrarily small for any $i<d$ and $\lambda_i$ arbitrarily large for any $i > 0$.

The quantities that are bounded, as Mark says,$\lambda_1 \lambda_2 \dots \lambda_r$, which is bounded above in this case by $1$, since $\lambda_{r+1},\dots, \lambda_d \geq \lambda_r \geq (\lambda_1\lambda_2 \dots \lambda_r)^{1/r}$. One can also consider $\lambda_{r+1} \lambda_{r+2} \dots \lambda_{d}$, which is bounded below by a constant - in this case, this is an equivalent statement, but for a general lattice it need not be.

It is generally a very good idea to test everything you want to prove for diagonal lattices. This is because of the general heuristic in lattice theory that any statement that holds for an arbitrary diagonal lattice should hold for an arbitrary lattice up to a constant.