Large sieve inequalities with power moduli and Waring's problem (Q6621275)

Let \(M\) be an integer, \(N\) be a positive integer, \(\{z_n\}\) be a sequence of complex numbers, \(\mathcal{S}\) be a finite set of real numbers, \(\delta\le 1\) be a positive real number, \(\lVert \alpha\rVert\) denotes the distance from a real number \(\alpha\) to its nearest integer, and \(e(\alpha)=e^{2\pi i\alpha}\). The set \(\mathcal{S}\) is \(\delta\)-spaced modulo 1 if \(\lVert \alpha-\beta\rVert\ge\delta\) whenever \(\alpha\) and \(\beta\) are distinct members of \(\mathcal{S}\). The general large sieve inequality says that if \(\mathcal{S}\) is \(\delta\)-spaced modulo 1, then\N\[\N\sum_{\alpha\in \mathcal{S}}\left|\sum_{n=M+1}^{M+N} z_n e(n\alpha)\right|^2\le (\delta^{-1}+N-1)\sum_{n=M+1}^{M+N}|z_n|^2.\N\]\NThe paper under review is about large sieve inequalities with \(k^{\textrm{th}}\)-power moduli, that are inequalities of the form \[\sum_{q=1}^Q\sum_{\substack{a=1\\\N\gcd(a,q)=1}}^{q^k}\left|\sum_{n=M+1}^{M+N} z_n e(n a/q^k)\right|^2\le \Delta_k(Q,N) \sum_{n=M+1}^{M+N}|z_n|^2,\] where \(\Delta_k(Q,N)\) is independent of \(M\) and \(\{z_n\}\). The authors let \(\Delta_k(Q,N)\) to be the infimum of such constants. The main results of the paper provide upper bounds on \(\Delta_k(Q,N)\). For all \(k\ge 2\), they show that\N\[\N\Delta_k(Q,N)\ll N^{\frac1{2}}Q^{k+\varepsilon},\N\]\Nand for all \(k\ge 3\),\N\[\N\Delta_k(Q,N)\ll N^{\frac{3}{4}}Q^{\frac{k}{2}+\frac{1}{4}+\frac{1}{2\sqrt{k}}+\varepsilon}.\N\]\NMoreover, they show that for any positive integer \(t\le k\), one has\N\[\N\Delta_k(Q,N)\ll N^{1-\frac{2}{t(t+1)}}Q^{1+\frac{4k-2t}{t(t+1)}+\varepsilon},\N\]\Nand as a corollary, the following bound\N\[\N\Delta_k(Q,N)\ll N^{1-\frac{2}{k(k+1)}}Q^{1+\frac{2}{k+1}+\varepsilon}.\N\]