Meaning of $\,r\bmod 1?\ $ [circle group $\Bbb R/\Bbb Z = $ reals $\!\bmod 1$]

Question

Let $A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}$ be a $2 \times 2$ matrix with integer entries and $\det(A) = \pm 1$. Consider the toral automorphism defined on 2-torus $f_A : T^2 \to T^2$, $$f_A (x, y) = (a x + b y \;\text{mod}\;1, c x + d y \;\text{mod}\; 1).$$ I don't understand what does it mean $a x + b y \;\text{mod}\;1$ and $c x + d y \;\text{mod}\; 1$. Can someone explain it to me?

Answer 1

In your context for a real number $r$ you have that $r \mod 1$ is the fractional part of $r$ that is the real number $r'$ in the interval $[0,1)$ such that $r-r'$ is an integer.

Or, more or less equivalently, it is the equivalence class of $r$ in $\mathbb{R}/\mathbb{Z}$, which contains $r$, that is, the set $r + \mathbb{Z}$.

I do not like the notation all that much but it is somewhat common. In a way it makes sense. It is the class of $r$ in $\mathbb{R}$ considered as a $\mathbb{Z}$ module with respect to the submodule $1 \cdot \mathbb{Z}$.

In this sense it matches the usage for integers.

Answer 2

This comes from identifying the torus with $S^1\times S^1$, and identifying the circle $S^1$ with $\Bbb R/\Bbb Z$.

A point on the circle is identified with a point on the real line where we forget all about everything to the left of the decimal point. Alternatively (and perhaps more correctly) we identify a point on the circle with a collection of points on the real line given by some $a\in \Bbb R$ along with all numbers $a+n$ for $n\in \Bbb Z$. Thus $0.5, 1.5, -2.5$ and $143.5$ all represent the same point on the circle, and we call that point on the circle "$0.5\!\!\!\mod 1$".

This is exactly analogous to how modular arithmetic works in number theory, only with real numbers instead of integers, so we steal notation from there.