Recall that the **$i$-th successive minimum** of $L\in \mathcal L$ (space of full rank lattices in $\mathbb R^d$), 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 unimodular lattice in $\mathbb R^d$. For $d\le 4$, it is possible to construct a basis $\{v_1,\dots, v_d\}$ of $\Lambda$ such that $\lambda_i(\Lambda)=\|v_i\|$ for any $i$.

If $d>4$, then the example $\frac{1}{2^{1/d}}L_0$, where $L_0:=Span_{\mathbb Z} \{e_1,\dots, e_{d-1},\frac{1}{2}(e_1+\cdots+e_d) \}$ shows it may be impossible to find such a basis. The coefficients $\frac{1}{2^{1/d}}$ is used to scale $L_0$ to a unimodular lattice.

It seems to me that such examples are really rare, but how rare?

Consider the set $S$ of unimodular lattices whose sucesssive minima cannot be attained by a basis of lattice. Recall the space of unimodular lattices are be identified with the homogeneous space $SL(d,\mathbb R)/SL(d,\mathbb Z)$. Can we see that $S$ is contained in (countable union) of submanifold/subvariety of dimension strictly lower than the full dimension? Or show it has Haar measure zero?

If it does not have measure zero, can we give an estimate of its measure?

(1) To begin with can anyone construct any examples different from this $\frac{1}{2^{1/d}}L_0$? (As Alison commented below small perturbation of this $\frac{1}{2^{1/d}}L_0$ example may work)

(2) Can any one construct any examples *significantly different* from this $\frac{1}{2^{1/d}}L_0$. For example, with $\lambda_1(\Lambda)$ close to zero? Such lattices are often called "unbounded".

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