Orthonormal Gabor systems and systems of translates

Show that $\mathcal{T}(f)$ and $\mathcal{T}(g)$ have the same closed spans if and only if $\widehat{f}(\xi ) = \alpha (\xi ) \widehat{g}(\xi )$ a.e., where $\alpha$ is 1 -periodic and $\left|\alpha (\xi )\right| = 1$ a.e.

Show that if $f$ and $g$ are each compactly supported, then $\overline{\operatorname {span}}(\mathcal{T}(f)) = \overline{\operatorname {span}}(\mathcal{T}(g))$ if and only $f = \alpha T_{k} g$ where $k \in \mathbb {Z}$ and $\alpha$ is a scalar with $\left|\alpha \right| = 1$.

Let $p(x) = \sum_{k = -N}^{N} c_{k} e^{2 \pi i k x}$ be a trigonometric polynomial. Show that if $\left|p\right| = 1$ a.e., then $p(x) = c_{n} e^{2 \pi i n x}$ for some $n$ between $-N$ and $N$.

Suppose that $g \in L^{2}(\mathbb {R})$ is compactly supported. Show that $\mathcal{G}(g, 1,1)$ is an orthonormal basis for $L^{2}(\mathbb {R})$ if and only if $\left|g\right| = \chi_{E}$ for some bounded set $E \subseteq \mathbb {R}$ that satisfies $\sum_{k \in \mathbb {Z}} \chi_{E}(x-k) = 1$ a.e.