TITLE: Need to bound a trigonometric sum
QUESTION [9 upvotes]: Let $\boldsymbol{\theta}=(\theta_1,\ldots,\theta_m)$ be a vector of real numbers in $[-\pi,\pi]$. For $t\ge 0$, define
$$ f(t,\boldsymbol{\theta}) = \binom{m+t-1}{t}^{-1}
 \sum_{j_1+\cdots+j_m=t} \exp(ij_1\theta_1+\cdots+ij_m\theta_m),$$
where the sum is over non-negative integers $j_1,\ldots,j_m$ with sum $t$.
Note that the number of terms in the sum is $\binom{m+t-1}{t}$, so
$|f(t,\boldsymbol{\theta})|\le 1$ with equality occurring when all the $\theta_j$s
are equal.
For a problem in asymptotic combinatorics, we need a bound on
$|f(t,\boldsymbol{\theta})|$ that decreases rapidly as the $\theta_j$s move apart and
is valid for all $\boldsymbol{\theta}$.
Surely this problem has been studied before?
Note that $\binom{m+t-1}{t}f(t,\boldsymbol{\theta})$ is the coefficient of $x^t$ in
$$\prod_{j=1}^m (1-xe^{i\theta_j})^{-1},$$
which suggests some sort of contour integral approach.

REPLY [2 votes]: We shall prove that 
$$ 
f(t,{\theta}) \le \frac{m-1}{t+m-1} \min_{1\le j,k \le m} \frac{1}{|\sin((\theta_j-\theta_k)/2)|}.
$$
This shows that if the angles are not too close to each other, then the sum does get 
small. 
Suppose without loss of generality that the minimum in our bound occurs for $\theta_1$ and $\theta_2$ 
(so these are the angles that are furthest apart).   Then writing $j_1+j_2 =\ell$ we have 
$$ 
|f(t,{ \theta})| \le \binom{m+t-1}{t}^{-1} \sum_{\ell=0}^{t} \sum_{j_3+\ldots+j_m=t-\ell} 
\Big| \sum_{j_1+j_2 =\ell} \exp(ij_1 \theta_1 + ij_2 \theta_2)\Big|. 
$$ 
Now the inner sum over $j_1$ (and $j_2=\ell -j_1$) is simply a geometric progression, and 
so 
\begin{align*} 
\Big| \sum_{j_1+j_2=\ell} \exp(ij_1 + i j_2 \theta_2) \Big| &= 
\Big| \sum_{j=0}^{\ell} \exp(ij (\theta_1-\theta_2))\Big| = 
\Big|\frac{\exp(i(\ell+1)(\theta_1-\theta_2))-1}{\exp(i(\theta_1-\theta_2))-1}\Big| 
\\
&\le \frac{2}{|\exp(i(\theta_1-\theta_2))-1|} = \frac{1}{|\sin((\theta_1-\theta_2)/2)|}.
\end{align*} 
Therefore 
$$ 
|f(t,{\theta})| \le \frac{1}{|\sin((\theta_1-\theta_2)/2)|}\binom{m+t-1}{t}^{-1} \sum_{\ell+j_3+\ldots+j_m=t} 1,
$$ 
which proves our claimed bound.
As the original question suggests, one would be able to prove better bounds 
using contour integrals.  The key would be to integrate on the circle centered at 
the origin and with radius $r=t/(m+t)$.  One should be able to get good bounds 
in terms of 
$$ 
\sum_{j,k} \sin^2 \Big(\frac{\theta_j-\theta_k}{2}\Big),
$$ 
but I have not worked this out carefully.