A fourth derivative test for exponential sums (Q2781950)

Let \(f:[1,M]\to \mathbb{R}\) be a function with four continuous derivatives that satisfies the van der Corput condition that \(\lambda \leq f^{(4)} (x) \ll \lambda\) for \(1\leq x \leq M\). Let \(S_M=\sum_{m\leq M} e(f(x))\). The classical van der Corput bound is \(S_M \ll M\lambda^{1/14}\), provided that \(M \gg \lambda^{-4/7}\). In this paper, the authors prove the estimate NEWLINE\[NEWLINES_M \ll_\varepsilon M^\varepsilon( M \lambda^{1/13} + \lambda^{-7/13}),NEWLINE\]NEWLINE and consequently, \(S_M \ll_\varepsilon M^{1+\varepsilon} \lambda^{1/13}\) provided \(M \gg \lambda^{-8/13}\). Their result can be phrased as saying that \((\theta+\varepsilon, 1-3\theta + \varepsilon)\) is an exponent pair with \(\theta=1/13\). This is weaker than a result of \textit{M. N. Huxley} and \textit{G. Kolesnik} [Exponential sums with a large second derivative, Number Theory, Turku, 1999, 131-143 (2001; Zbl 0966.11035)], which gives \((\theta+\varepsilon, 1-3\theta + \varepsilon)\) with \(\theta=1/12.84 \dots\). However, the proof given here is simpler i.o. and is stronger for short exponential sums. NEWLINENEWLINENEWLINEIn the proof, the authors begin by applying the one-dimensional van der Corput process \(A\) and the two-dimensional process \(A\times A\). They then shift the main variable to produce a new variable. The resulting quadruple exponential sum is estimated via the Bombieri-Iwaniec double large sieve. The initial problem is reduced to counting the number of solutions of a particular Diophantine system.