A lattice $\Gamma \subset \mathbb{R}^d$ is said to be rational if for any two vectors $\gamma_1, \gamma_2 \in \Gamma$, their inner product satisfies the relation

$$\displaystyle \langle\gamma_1, \gamma_2 \rangle = \beta_{\gamma}r_{12},$$

where $\beta_\Gamma \neq 0$ is a real-valued constant independent of both $\gamma_1$ and $\gamma_2$, and $r_{12} = r_{21}$ is an integer. Otherwise, we say that the lattice is irrational.

I'd like to talk about the quotient $\mathbb{R}^d / \Gamma$. If $\Gamma$ is a rational lattice, say, $\Gamma = \mathbb{Z}^d$, then we have $\mathbb{R}^d / \Gamma = (0,1)^d,$ a torus.

My question is, what happens if $\Gamma$ is irrational? Do we still obtain a torus? Could someone give me an example of an irrational lattice and what the resulting quotient looks like?


The quotient of $\mathbb R^d$ by any full-rank lattice $\Gamma$ is a torus, no matter what are the properties of $\Gamma$. This is because any two lattices are related by an invertible matrix, the action of this matrix on the quotient space will give an homeomorphism between the tori.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.