An exact renormalization formula for Gaussian exponential sums and applications (Q2892853)

Let \(\{a,b\} \in (0,1)\times (-\dfrac12,\dfrac12]\). The authors study the Gaussian exponential sum NEWLINE\[NEWLINE S(N,a,b)=\sum\limits_{0\leq n\leq N-1}\exp \left\{ 2\pi i(-\frac{an^2}2+nb)\right\},\quad n\in \mathbb N. NEWLINE\]NEWLINE They prove a renormalization formula expressing \(S(N,a,b)\) in terms of the sum \(S(N_1,a_1,b_1)\) containing a smaller number of terms, with an explicit remainder term (the first results of this kind were proved by \textit{G. H. Hardy} and \textit{J. E. Littlewood} [Acta Math. 37, 155--191 (1914), 37, 193--239 (1914); JFM 45.0305.03]). This formula implies precise growth estimates for \(|S(N,a,b)|\), as \(N\to \infty\).