Let $p$ be a prime, $a \neq 0$ an integer, let $M,N \gg 1$ and let $\psi,\eta$ be some fixed Schwartz functions. Would you know of any references in the literature where upper bounds for sums such as $$ \sum_{m,n\geq 1} \psi(m/M)\eta(n/N) e_p(amn) $$ were obtained?
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This should follow by approximating $a/p$ by $a/q+\beta$ with $q^2\ll MN$ and $q|\beta|\ll 1/\sqrt{MN}$. For simplicity, I’ll suppose $\beta=0$ (this will be the Archimedean analogue of the $q$-treatment). Applying Poisson summation in $m$, $n$ should give an asymptotic for $q=o(\sqrt{MN})$ and a sharp upper bound of $\sqrt{MN}$ if $q\gg\sqrt{MN}$. This will hold more generally with $q(1+MN|\beta|)$ in place of $q$ (some exponential sum evaluations will be accompanied with stationary phase now).
