On Waring’s problem: Some refinements. (English) Zbl 0736.11058
In this interesting paper various refinements connected with the first author’s new iterative method [Acta Math. 162, 1-71 (1989; Zbl 0665.10033){]} in the theory of Waring’s problem are discussed. One point at issue is a technique introduced by the first author [J. Reine Angew. Math. 365, 122-170 (1986; Zbl 0574.10046){]} which can be used to avoid the standard use of Weyl’s inequality in the final step of the iteration process. Instead of using Weyl’s bound for \(k\)-th powers one bounds an allied sum with \((k-1)-th\) powers. In certain circumstances this reduces an upper bound \(G(k)\leq H(k)\) to \(G(k)\leq H(k)-1\). In this way it is shown that \(G(5)\leq18\) and \(G(6)\leq28\). The reviewer has also shown \(G(5)\leq18\) based on the same observation [Proc. Lond. Math. Soc., III. Ser. 61, 457-479 (1990; Zbl 0677.10036){]} but the method used here has the advantage that it produces an essentially best possibly lower bound for the number of representations by sums of 18 fifth powers.
The second innovation improves on Vaughan’s new iterative method itself. One of the ingredients of this technique is a mean value bound for a “differenced Weyl sum”, \[ F(\alpha)=\sum_{x,h,m}e(\alpha m^{- k}((x+hm^ k)^ k-(x-hm^ k)^ k)) \] (\(x,h,m\) in certain intervals). Bounds for \(\int^ 1_ 0| F(\alpha)|^{2s}d\alpha\) were deduced by the first author (loc. cit.) by a Hua lemma type argument. For larger \(k\), this is now improved by invoking bounds from Vinogradov’s mean-value theorem.
The exponential sums \(\sum_{x\in{\mathcal A}(P;R)}e(\alpha x^ k)\) with \({\mathcal A}(P,R)=\{x\leq P: p\mid x\Rightarrow p\leq R\}\) play a crucial role in recent advances with Waring’s problem. Two new upper bounds of a major arc type are obtained. The first one is based on the large sieve, and gives a bound of the type \(\ll Pq^{\varepsilon-1/2k}\) if \(\alpha\) is close to \(a/q\) with \((a,q)=1\) and \(q\) not too large, very roughly speaking. The second one uses a recent result of E. Fouvry and G. Tenenbaum [Proc. Lond. Math. Soc., III. Ser. 63, 449-494 (1991){]} on the distribution of \({\mathcal A}(P,R)\) in arithmetic progressions and gives, again roughly speaking, the essentially best possible upper bound \(Pq^{\varepsilon-1/k}\) if \(q\leq(\log P)^ A\) for some \(A>0\). These results will play their role in further work on the subject, so the reviewer suspects.
When combined these ideas lead to new upper bounds for \(G(k)\) when \(k\geq10\). For comparison we quote the results \(G(15)\leq168\), \(G(20)\leq243\), and the previous bounds due to Vaughan (loc. cit.) \(G(15)\leq171\), \(G(20)\leq248.\)
The second innovation improves on Vaughan’s new iterative method itself. One of the ingredients of this technique is a mean value bound for a “differenced Weyl sum”, \[ F(\alpha)=\sum_{x,h,m}e(\alpha m^{- k}((x+hm^ k)^ k-(x-hm^ k)^ k)) \] (\(x,h,m\) in certain intervals). Bounds for \(\int^ 1_ 0| F(\alpha)|^{2s}d\alpha\) were deduced by the first author (loc. cit.) by a Hua lemma type argument. For larger \(k\), this is now improved by invoking bounds from Vinogradov’s mean-value theorem.
The exponential sums \(\sum_{x\in{\mathcal A}(P;R)}e(\alpha x^ k)\) with \({\mathcal A}(P,R)=\{x\leq P: p\mid x\Rightarrow p\leq R\}\) play a crucial role in recent advances with Waring’s problem. Two new upper bounds of a major arc type are obtained. The first one is based on the large sieve, and gives a bound of the type \(\ll Pq^{\varepsilon-1/2k}\) if \(\alpha\) is close to \(a/q\) with \((a,q)=1\) and \(q\) not too large, very roughly speaking. The second one uses a recent result of E. Fouvry and G. Tenenbaum [Proc. Lond. Math. Soc., III. Ser. 63, 449-494 (1991){]} on the distribution of \({\mathcal A}(P,R)\) in arithmetic progressions and gives, again roughly speaking, the essentially best possible upper bound \(Pq^{\varepsilon-1/k}\) if \(q\leq(\log P)^ A\) for some \(A>0\). These results will play their role in further work on the subject, so the reviewer suspects.
When combined these ideas lead to new upper bounds for \(G(k)\) when \(k\geq10\). For comparison we quote the results \(G(15)\leq168\), \(G(20)\leq243\), and the previous bounds due to Vaughan (loc. cit.) \(G(15)\leq171\), \(G(20)\leq248.\)
Reviewer: J.Brüdern (Göttingen)
