A uniform estimate for simple stationary phase integrals (Q2716950)

In using van der Corput's method to estimate exponential sums [see Expo. Math. 17, 207-232 (1999; Zbl 0969.11028) by the same authors], one often encounters exponential integrals of the type NEWLINE\[NEWLINEJ= \int_a^b e(f(x)) dx,NEWLINE\]NEWLINE where \(f\) is a smooth real-valued function. Under suitable circumstances one has NEWLINE\[NEWLINEJ= e^{i\pi/4} \frac{e(f(c))} {\sqrt{f''(c)}}+ E,NEWLINE\]NEWLINE where \(f'(c)=0\) and \(E\) is a small error. The classical bound for \(E\) takes the form \(E\ll MT^{-1/2}\), when the derivatives \(f^{(k)}\) behave like \(TM^{-k}\). The first result gives an expression for \(E\) involving subsidiary explicit terms and an error \(O(MT^{-3/2})\). This may be seen as a more precise version of a result of \textit{M. N. Huxley} [Glasg. Math. J. 36, 355-362 (1994; Zbl 0824.11047)]. The paper then proceeds to estimate an average \(\sum_{h\leq H} E_h\), formed by replacing \(f\) by a sequence of functions \(f_h\) depending smoothly on the parameter \(h\). It is shown that NEWLINE\[NEWLINE\sum_{h\leq H}E_h\ll \frac{M}{T^{1/2}}+ \frac{HM}{T^{3/2}}+ \min\Biggl\{ \frac{HM}{T}, \frac{M}{T^{1/2} \|\alpha\|} \Biggr\}\log H,NEWLINE\]NEWLINE for an explicitly described real \(\alpha\), depending on \(a\), \(b\) and the functions \(f_h\). In suitable circumstances this is significantly sharper than previous bounds.