Mean value estimates for odd cubic Weyl sums

Wooley, Trevor D.

doi:10.1112/blms/bdv066

Cited by 19 publications

(26 citation statements)

References 11 publications

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“…This mean value played a critical role in his approach to Vinogradov's mean value theorem for small degrees (see [, Chapter 5]). Thus, Hua obtained the bounds

S_{3} ⩽ 10, S_{4} ⩽ 32, S_{5} ⩽ 86, \dots .

More recently, as a consequence of progress on Vinogradov's mean value theorem stemming from the efficient differencing method, the author obtained the bounds

S_{k} ⩽ 2 k^{2} - 2 k

(see [, Theorem 11.6]) and

S_{3} ⩽ 9

(see [, Theorem 1.1]).…”

Section: Analogues Of Hua's Lemma and Waring's Problemmentioning

confidence: 99%

Nested efficient congruencing and relatives of Vinogradov's mean value theorem

Wooley

2018

Proc. London Math. Soc.

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We apply a nested variant of multigrade efficient congruencing to estimate mean values related to that of Vinogradov. We show that when φj∈Zfalse[tfalse] (1⩽j⩽k) is a system of polynomials with non‐vanishing Wronskian, and s⩽k(k+1)/2, then for all complex sequences (frakturan), and for each ε>0, one has 0true∫[0,1)k0true∑|n|⩽Xfrakturanefalse(α1φ1(n)+⋯+αkφk(n)false)2sdα≪Xε0true∑|n|⩽X|frakturan|2s.As a special case of this result, we confirm the main conjecture in Vinogradov's mean value theorem for all exponents k, recovering the recent conclusions of the author (for k=3) and Bourgain, Demeter and Guth (for k⩾4). In contrast with the l2‐decoupling method of the latter authors, we make no use of multilinear Kakeya estimates, and thus our methods are of sufficient flexibility to be applicable in algebraic number fields, and in function fields. We outline such extensions.

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“…This mean value played a critical role in his approach to Vinogradov's mean value theorem for small degrees (see [, Chapter 5]). Thus, Hua obtained the bounds

S_{3} ⩽ 10, S_{4} ⩽ 32, S_{5} ⩽ 86, \dots .

More recently, as a consequence of progress on Vinogradov's mean value theorem stemming from the efficient differencing method, the author obtained the bounds

S_{k} ⩽ 2 k^{2} - 2 k

(see [, Theorem 11.6]) and

S_{3} ⩽ 9

(see [, Theorem 1.1]).…”

Section: Analogues Of Hua's Lemma and Waring's Problemmentioning

confidence: 99%

Nested efficient congruencing and relatives of Vinogradov's mean value theorem

Wooley

2018

Proc. London Math. Soc.

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126

141

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“…where, as usual, we write e(z) for e 2πiz . However, these applications require somewhat elaborate arguments that preclude their inclusion in this paper, and so we defer accounts of such developments to forthcoming papers [12,13] elsewhere. The proof of the cubic case of the main conjecture seems worthy in its own right as the highlight of this memoir.…”

Section: Introductionmentioning

confidence: 99%

The cubic case of the main conjecture in Vinogradov's mean value theorem

Wooley

2016

Advances in Mathematics

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We apply a variant of the multigrade efficient congruencing method to estimate Vinogradov's integral of degree 3 for moments of order 2s, establishing strongly diagonal behaviour for 1 s 6. Consequently, the main conjecture is now known to hold for the first time in a case of degree exceeding 2. s (1 j k),(1.1)with 1 x i , y i X (1 i s). The main conjecture in Vinogradov's mean value theorem asserts that for each ε > 0, one hasan estimate that, but for the presence of the factor X ε , would be best possible (see [5, equation (7.4)]). Despite eighty years of intense investigation, such an estimate has been established only in two cases, namely the (trivial) linear case with k = 1, and the quadratic case with k = 2 in which the elementary theory of quadratic forms can be brought to bear. Our goal in this paper is the first proof of the main conjecture (1.2) in a case with k > 2.Theorem 1.1. For each ε > 0, one has J s,3 (X) ≪ X ε (X s + X 2s−6 ).The estimate for J s,3 (X) recorded in this theorem, which establishes the main conjecture in Vinogradov's mean value theorem in the cubic case k = 3, goes substantially beyond the estimates available hitherto. By means of Newton's formulae concerning the roots of polynomials, it is apparent that J s,3 (X) = s!X s + O(X s−1 ) for 1 s 3, since the solutions of (1.1) are then simply the diagonal ones with {x 1 , . . . , x s } = {y 1 , . . . , y s }. Moreover, from [6, Theorem 1.5] one has J 4,3 (X) = 4!X 4 + O(X 10/3 (log 2X) 35 ).These estimates confirm (1.2) for 1 s 4 in a particularly strong form when k = 3, though in the latter range the estimate (1.2) has been known since at

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“…The reader should have no difficulty, however, in either adapting the methods underlying these cited bounds, or indeed deriving the stated results through application of the triangle inequality. Improved bounds for the former mean value are the subject of , whilst the second is handled more precisely in and . Thus, we obtain the estimate