Local mean value estimates for Weyl sums (Q6653230)

For positive integers \(d\) and \(N\), a sequence of complex weights \(\boldsymbol{a}= (a_n)_{n=1}^N\), and a vector \(\boldsymbol{x}=(x_1,\ldots, x_d)\in\mathsf{T}_d:=[0,1)^d\), we define the Weyl sums \(S_d(\boldsymbol{x}; \boldsymbol{a}, N)\) as follows\N\[\NS_d(\boldsymbol{x}; \boldsymbol{a}, N) = \sum_{n=1}^N a_n \mathrm{e}(x_1n + \ldots + x_d n^d).\N\]\NAs usual, \(\mathrm{e}(z) = \exp(2\pi i z)\). Letting \(\boldsymbol{1}\) to coincide with \(a_n=1\), the special case\N\[\NS_d(\boldsymbol{x}; N):=S_d(\boldsymbol{x}; \boldsymbol{1}, N)\N\]\Nis associated to Vinogradov's mean value theorem, and is well studied on average as \(\boldsymbol{x}\) ranges over the unit hypercube \(\mathsf{T}_d\). However, the pointwise size of \(S_d(\boldsymbol{x}; N)\) for any fixed point \(\boldsymbol{x}\) is still far from the conjectured bounds. The authors' work in the present paper is an attempt to interpolate between our almost complete understanding of mean values of \(S_d(\boldsymbol{x}; N)\) and our deficient understanding of the pointwise behaviour of these sums. To do this, they investigate \(S_d(\boldsymbol{x}; N)\) and related exponential sums as \(\boldsymbol{x}\) ranges over small boxes of the form \(\boldsymbol{\xi}+ [0,\delta]^d\), where \(0<\delta\le 1\) and \(\boldsymbol{\xi} \in \mathsf{T}_d\). More precisely, letting\N\[\NI_{s, d}(\delta, \boldsymbol{\xi}; \boldsymbol{a}, N)=\int_{\boldsymbol{\xi}+ [0,\delta]^d} \left| S_d(\boldsymbol{x}; \boldsymbol{a}, N)\right |^{2s} d\boldsymbol{x},\N\]\Nand \(I_{s, d}^{(0)}(\delta; \boldsymbol{a}, N):= I_{s, d}(\delta, \boldsymbol{0}; \boldsymbol{a}, N)\), they obtain several bounds on \(I_{s, d}^{(0)}(\delta; \boldsymbol{a}, N)\) in small and large ranges for \(s\), determined according to \(s(d)=d(d+1)/2\). The authors end the paper by some notes, including the note that their results have consequences for solutions of Schrödinger equations \(2\pi u_t+iu_{xx}=0\) over short intervals, which models the behaviour of quantum mechanical particles.