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.