Bound of inf of finite Dirichlet series $\sum e^{i\alpha_j t}.$

Question

In a short note by S. Rudolfer and W.K. Hayman (1980), there is a sketch of a proof for an upper bound of $\mu =\inf |P(t)|$ where $$P(t) = \sum_{j=1}^n e^{i\alpha_j t}$$ and $\alpha_j \neq \alpha_k$ for $j\neq k$. There are a couple points of the argument that I can't puzzle through. They assert that $\mu \leq \sqrt{n-1}.$

Here is the argument: Note that $|P(t)|^2 = n + 2\sum_{j\neq k} \cos((\alpha_j-\alpha_k)t).$ If $w(t) = |P|^2-n \geq -c$, then let

$$h(t) = c+w(t) = 2\sum b_k \cos \delta_k t$$

where the $b_k$ are positive integers. (No limits indicated on this last summation.) Then

$$b_k = \lim_{T\to \infty}\frac{1}{2T}\int_{-T}^{T} h(t)\cos(\delta_k t) \; dt \leq \lim_{T\to \infty}\frac{1}{2T}\int_{-T}^{T} h(t) \; dt = c.$$

They conclude that $c\geq 1$.

My first problem is, "What are the $\delta_k$'s? At first I thought they were just the collection of the various $(\alpha_j -\alpha_k)$'s but now I don't think so. Are they just saying, "Take the Fourier series of $h(t)$?"

My second problem is, I'm not familiar with any sort of Fourier series where we take the limit as $T\to \infty.$ Just a reference for such a thing would suffice, I think.

Third, having shown $c\geq 1$, how does that get us from $|P|^2 - n \geq -c$ to $\inf |P| \leq \sqrt{n-1}$.

Answer 1

1. Among $\alpha_i-\alpha_j$ there could be same numbers. I think that $\delta_k$'s denote different numbers in the set $\{\alpha_i-\alpha_j\}$. Therefore, $b_k$ is exactly the number of repetition of each $\delta_k$ in the set.
2. No need to go for Fourier transforms etc. It is simple integration: calculate $$ \frac{1}{2T}\int_{-T}^T\cos\alpha t\cdot\cos\beta t\,dt $$ when $\alpha\ne\beta$ and when $\alpha=\beta$ and then look what happens when $T\to+\infty$. You will see that cosines with different frequencies are kind of "orthonormal" in this sense.
3. Since $-c$ can be any lower bound for $w$, we can take it to be the exact lower bound, i.e. $-c=\inf(|P|^2-n)$. It makes $$ \inf|P|^2=n-c\le n-1. $$