Edit : no answer, no comment ... let's try with a chocolate bar.

Let $L$ be a maximal integral lattice in the euclidean $(\mathbf R^{8m},q)$ (thus the associated bilinear form $b(u,v)=q(u+v)-q(u)-q(v)$, once restricted to $L$, takes values in $2\mathbf Z$ and has discriminant $1$).

One may define the following :

$$X(L):=\bigcup_{s>0}\bigcup_{q(x)\equiv 1\text{ mod } s} B(\frac{x}{s},\frac{1}{s}) $$

($s$ runs over the natural number, and $B(x,\delta)$ is the open ball of radius $\delta$ centered at $x$).

Question 1 : Is $X(L)$ connected ?

Wouldn't there be the restriction on $q(x)$, this would be true, and in fact, $X(L)$ would be the whole $\mathbf R^{8m}$ (by Minkowski's convex body theorem).

I have checked that

• it is the case for $m=1,2,3$, but have no clue for the general case,

• the answer does not depend of the $L$ you choose (as long as it is maximal integral),

• the centers of the balls form a dense subset of $\mathbf R^{8m}$.

In fact, the first two of these remarks follow from the equivalence of the question with the following

Question 2 : Is the Coxeter graph for the (unique up to isomophism) maximal integral lattice (often denoted by $\text{II}_{1,8m+1}$) on the Lorentzian $\mathbf R^{8m+2}$ connected ?

Let $L$ be a maximal integral lattice in the euclidean $(\mathbf R^{8m},q)$ (thus the associated bilinear form $b(u,v)=q(u+v)-q(u)-q(v)$, once restricted to $L$, takes values in $2\mathbf Z$ and has discriminant $1$).

One may define the following :

$$X(L):=\bigcup_{s>0}\bigcup_{q(x)\equiv 1\text{ mod } s} B(\frac{x}{s},\frac{1}{s}) $$

($s$ runs over the natural number, and $B(x,\delta)$ is the open ball of radius $\delta$ centered at $x$).

Question 1 : Is $X(L)$ connected ?

Wouldn't there be the restriction on $q(x)$, this would be true, and in fact, $X(L)$ would be the whole $\mathbf R^{8m}$ (by Minkowski's convex body theorem).

I have checked that

• it is the case for $m=1,2,3$, but have no clue for the general case,

• the answer does not depend of the $L$ you choose (as long as it is maximal integral),

• the centers of the balls form a dense subset of $\mathbf R^{8m}$.

In fact, the first two of these remarks follow from the equivalence of the question with the following

Question 2 : Is the Coxeter graph for the (unique up to isomophism) maximal integral lattice (often denoted by $\text{II}_{1,8m+1}$) on the Lorentzian $\mathbf R^{8m+2}$ connected ?

Let $L$ be a maximal integral lattice in the euclidean $(\mathbf R^{8m},q)$ (thus the associated bilinear form $b(u,v)=q(u+v)-q(u)-q(v)$, once restricted to $L$, takes values in $2\mathbf Z$ and has discriminant $1$).

One may define the following :

$$X(L):=\bigcup_{s>0}\bigcup_{q(x)\equiv 1\text{ mod } s} B(\frac{x}{s},\frac{1}{s}) $$

($s$ runs over the natural number, and $B(x,\delta)$ is the open ball of radius $\delta$ centered at $x$).

Question 1 : Is $X(L)$ connected ?

Wouldn't there be the restriction on $q(x)$, this would be true, and in fact, $X(L)$ would be the whole $\mathbf R^{8m}$ (by Minkowski's convex body theorem).

I have checked that

• it is the case for $m=1,2,3$, but have no clue for the general case,

• the answer does not depend of the $L$ you choose (as long as it is maximal integral),

• the centers of the balls form a dense subset of $\mathbf R^{8m}$.

In fact, the first two of these remarks follow from the equivalence of the question with the following

Question 2 : Is the Coxeter graph for the (unique up to isomophism) maximal integral lattice (often denoted by $\text{II}_{1,8m+1}$) on the Lorentzian $\mathbf R^{8m+2}$ connected ?

Edit : no answer, no comment ... let's try with a chocolate bar.

Let $L$ be a maximal integral lattice in the euclidean $(\mathbf R^{8m},q)$ (thus the associated bilinear form $b(u,v)=q(u+v)-q(u)-q(v)$, once restricted to $L$, takes values in $2\mathbf Z$ and has discriminant $1$).

One may define the following :

$$X(L):=\bigcup_{s>0}\bigcup_{q(x)\equiv 1\text{ mod } s} B(\frac{x}{s},\frac{1}{s}) $$

($s$ runs over the natural number, and $B(x,\delta)$ is the open ball of radius $\delta$ centered at $x$).

Question 1 : Is $X(L)$ connected ?

Wouldn't there be the restriction on $q(x)$, this would be true, and in fact, $X(L)$ would be the whole $\mathbf R^{8m}$ (by Minkowski's convex body theorem).

I have checked that

• it is the case for $m=1,2,3$, but have no clue for the general case,

• the answer does not depend of the $L$ you choose (as long as it is maximal integral),

• the centers of the balls form a dense subset of $\mathbf R^{8m}$.

In fact, the first two of these remarks follow from the equivalence of the question with the following

Question 2 : Is the Coxeter graph for the (unique up to isomophism) maximal integral lattice (often denoted by $\text{II}_{1,8m+1}$) on the Lorentzian $\mathbf R^{8m+2}$ connected ?