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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 two 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.

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.

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 \Lambda\}, \\ B_i := \sup \{\lambda_i(L):L\in \Lambda\}. \end{gather*}

I know two 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.

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 two 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.

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 \Lambda\}, \\ B_i := \sup \{\lambda_i(L):L\in \Lambda\}. \end{gather*}

I know two of them from Minkowski's second convex body theorem, namely $B_1 < \infty$ and $A_d >0$. In addition, $A_1=0$ via the following example $ \begin{bmatrix} e^{\frac{d-1}{d}t}I_{d-1} & 0 \\ 0 & e^{-\frac{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.

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 \Lambda\}, \\ B_i := \sup \{\lambda_i(L):L\in \Lambda\}. \end{gather*}

I know two 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.