I was wondering what is the 2 dimensional Lebesgue measure of this set
$\Gamma^2 := \{ (\{at\},\{bt\}) \; : \; t \in \mathbb{R} \} \subset \mathbb{R}^2$
Where $\{x\}$ denotes the fractional part of ,$x$ (for example $\{\pi\} = 0.1415\dots$ and $a,b$ are two irrational numbers such that $a/b$ is irrational
Let $I_{j, k}^2 = [j, j + 1) \times[k, k+1)$ for $j, k \in \mathbb Z$. The set $T_{I_{j, k}^2} = \{ t \in \mathbb R \colon (at, bt) \in I_{j, k}^2 \}$ is an interval (or empty) in $\mathbb R$. If $\gamma$ is the map such that $\gamma(t) = \left( at, bt \right)$, then $\gamma(T_{I_k^2}) \subset I_{j, k}^2$ is a line, and we know that $\lambda^2\left(\gamma(T_{I_{j, k}^2})\right) = 0$.
If $\gamma^*$ is the map such that $\gamma^*(t) = \left( \{at\}, \{bt\} \right)$, then $\lambda^2\left(\gamma^*(T_{I_{j, k}^2})\right) = \lambda^2\left(\gamma(T_{I_{j, k}^2})\right) = 0$ by translation invariance of the Lebesgue measure. Therefore
\begin{align*} \lambda^2\left(\bigcup_{j = -\infty}^{\infty} \bigcup_{k = -\infty}^{\infty} \gamma(T_{I_{j, k}^2})\right) &= \sum_{j = -\infty}^{\infty} \sum_{k = -\infty}^{\infty} \lambda^2\left( \gamma(T_{I_{j, k}^2})\right)\\ &= \underbrace{\sum_{j = -\infty}^{\infty} \sum_{k = -\infty}^{\infty} \lambda^2\left( \gamma^*(T_{I_{j, k}^2})\right)}_{= 0}\\ &\geq \lambda^2\left(\bigcup_{j = -\infty}^{\infty} \bigcup_{k = -\infty}^{\infty} \gamma^*(T_{I_{j, k}^2})\right)\\ &= \lambda^2\left(\gamma^*(\mathbb R)\right)\\ &= \lambda^2\left(\Gamma^2\right) \end{align*}