A new bound for the large sieve inequality with power moduli (Q2880331)

The classical large sieve is the inequality NEWLINE\[NEWLINE \sum\limits_{q\leq Q} \sum\limits_{_{\substack{ a=1\\ (a,q)=1}}}^q \left| \sum\limits_{M<n\leq M+N} a_n e\left(n\cdot \frac{a}{q}\right)\right|^2 \leq \left(N+Q^2\right) \sum\limits_{M<n\leq M+N} |a_n|^2 NEWLINE\]NEWLINE for \(N,Q\in \mathbb{N}\). L. Zhao and the reviewer considered the situation when the moduli \(q\) are restricted to a sparse set, both independently and in joint work. The paper under review improves their results in [\textit{S. Baier} and \textit{L. Zhao}, Int. J. Number Theory 1, 265--279 (2005; Zbl 1083.11060)] on the large sieve with power moduli. The author proves that NEWLINE\[NEWLINE \sum\limits_{q\leq Q} \sum\limits_{_{\substack{ a=1\\ (a,q)=1}}}^{q^k} \left| \sum\limits_{M<n\leq M+N} a_n e\left(n\cdot \frac{a}{q^k}\right)\right|^2 \ll_{k,\varepsilon} (NQ)^\varepsilon\left(Q^{k+1}+Q^{1-\delta}+Q^{1+k\delta}N^{1-\delta}\right) NEWLINE\]NEWLINE for all \(N,Q,k \in \mathbb{N}\), where \(\delta:=(2k(k-1))^{-1}\). The main novelty in her method is to use Wooley's recent bound in [\textit{T. Wooley}, Ann. Math. (2) 175, No. 3, 1575--1627 (2012; Zbl 1267.11105)] for exponential sums with polynomial amplitude function, achieved via efficient congruencing.