modulus of the function does not exceed 1.

We use counting measure on ${\mathbb{Z}}$, so that for $f,g :{\mathbb{Z}}\to {\mathbb{C}}$, we have $${\left\| f\right\|}_{\ell^p} := {\biggl( \sum_x |f(x)|^p \biggr)}^{{\frac{1}{p}}}, \ {\left\langlef,g\right\rangle} := \sum_x f(x)\overline{g(x)},\ \text{and}\ (f*g)(x) = \sum_y f(y)g(x-y).$$ Any sum of the form $\sum_x$ is to be interpreted as a sum over ${\mathbb{Z}}$. The *support* of $f$ is the set ${\mathrm{supp}}(f) := {\left\{x \in {\mathbb{Z}}: f(x) \neq 0\right\}}$. We write ${\left\| f\right\|}_\infty$ for $\sup_{x \in {\mathbb{Z}}} |f(x)|$.

We use Haar probability measure on ${\mathbb{T}}:= {\mathbb{R}}/{\mathbb{Z}}$, so that for measurable $F : {\mathbb{T}}\to {\mathbb{C}}$, we have $${\left\| F\right\|}_{L^p} := {\biggl( \int_{\mathbb{T}}|F(\alpha)|^pd\alpha \biggr)}^{{\frac{1}{p}}} = {\biggl( \int_0^1 |F(\alpha)|^pd\alpha \biggr)}^{{\frac{1}{p}}}.$$ We write ${\left\| \alpha\right\|}_{\mathbb{T}}$ for the distance from $\alpha \in {\mathbb{R}}$ to the nearest integer $
\min_{n \in {\mathbb{Z}}} |\alpha - n|.
$ This remains well-defined on ${\mathbb{T}}$.

We define the Fourier transform of $f : {\mathbb{Z}}\to {\mathbb{C}}$ by $$\label{Fourier transform}
\hat{f}(\alpha) := \sum_x f(x) e(\alpha x) \qquad (\alpha \in {\mathbb{T}}),$$ when this makes sense. Here $e(\alpha)$ stands for $e^{2\pi i \alpha}$.

For a finite set $S$ and function $f:S\to{\mathbb{C}}$, denote the average of $f$ over $S$ by $${\mathbb{E}}_{s\in S}f(s):=\frac{1}{|S|}\sum_{s\in S}f(s).$$

If $\|\cdot\|$ is a seminorm on an inner product space, recall that 