a) Let $\beta \in \mathbb{R}$ and let $g$ be the function defined by $g(t) = \mathrm{e}^{\mathrm{i}\beta t}$ for all $t \in \mathbb{R}$. Determine the codimension of $N_{g}$ in $L^{1}(\mathbb{R})$.\\
b) Let $n$ be a natural number. Show that there exists a function $g$ in $C_{b}(\mathbb{R})$ such that $N_{g}$ has codimension $n$ in $L^{1}(\mathbb{R})$.