New estimates for mean values of Weyl sums. (English) Zbl 0821.11050

Let \(f(x)\) be a non-zero polynomial of degree \(k\) with integer coefficients. This paper is concerned with estimating the mean-value \[ I_ s (P)= \int_ 0^ 1 | \sum_{n\leq P} \exp\{ 2\pi i\alpha f(n)\} |^{2s} d\alpha. \] The estimate is given in terms of the exponent in Vinogradov’s mean-value theorem. Thus one writes \(J_ s (P)\) for the number of solutions of the simultaneous equations \(\sum_{i=1}^ s x_ i^ j= \sum_{i=1}^ s y_ i^ j\) \((1\leq j\leq k)\) in positive integers \(x_ i, y_ i\leq P\). The standard estimate for \(I_ s (P)\) states that if \[ J_ s (P)\;\ll\;P^{2s- k(k+ 1)/2+ \eta (s,k)}, \] then \(I_ s(P)\ll P^{2s-k +\eta (s,k)}\). The proof of this is very easy. The present paper makes a significant advance by establishing the estimate \[ I_ s (P)\;\ll\;P^{2s- k+\eta (s- m(m- 1)/2, k)/m} \] for any positive integer \(m\leq k\). Using T. D. Wooley’s bound [Mathematika 39, 379-399 (1992; Zbl 0769.11036)] for \(\eta (s,k)\) one finds that the optimal choice for \(m\) is around \(k/ \sqrt {2}\), which produces an exponent something like \(2s- k+ {{\sqrt{2e}} \over k} \eta (s,k)\). One is then able to establish the asymptotic formula in Waring’s problem for \(k^ 2 (\log k+\log \log k+ O(1))\) or more \(k\)-th powers. This halves the previous lower bound, due to Wooley (loc. cit.). The proof of the main estimate uses an iterative scheme of the same general nature as that used by Wooley.


11P05 Waring’s problem and variants
11L15 Weyl sums


Zbl 0769.11036
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