Near-Optimal Mean Value Estimates for Multidimensional Weyl Sums

Parsell, Scott T.; Prendiville, Sean; Wooley, Trevor D.

doi:10.1007/s00039-013-0242-7

Cited by 38 publications

(85 citation statements)

References 17 publications

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“…We consequently propose to expand no further on this subject, leaving the reader to complete the routine exercises needed for its proof, and to apply [16] as the necessary framework.…”

Section: Wider Applications Of Weighted Efficient Congruencingmentioning

confidence: 99%

Discrete Fourier restriction via Efficient Congruencing

Wooley¹

2016

Int Math Res Notices

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General rightsThis document is made available in accordance with publisher policies. Please cite only the published version using the reference above. Full terms of use are available: http://www.bristol.ac.uk/pure/about/ebr-termsAbstract. We show that whenever s > k(k + 1), then for any complex sequence (a n ) n∈Z , one hasBounds for the constant in the associated periodic Strichartz inequality from L 2s to l 2 of the conjectured order of magnitude follow, and likewise for the constant in the discrete Fourier restriction problem from l 2 to L s , where s = 2s/(2s − 1). These bounds are obtained by generalising the efficient congruencing method from Vinogradov's mean value theorem to the present setting, introducing tools of wider application into the subject.

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“…We consequently propose to expand no further on this subject, leaving the reader to complete the routine exercises needed for its proof, and to apply [16] as the necessary framework.…”

Section: Wider Applications Of Weighted Efficient Congruencingmentioning

confidence: 99%

Discrete Fourier restriction via Efficient Congruencing

Wooley¹

2016

Int Math Res Notices

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“…For later refinements of the Vinogradov mean value theorem see, e.g., [23], [24], [25], and the discussion and references therein. Since there are numerous applications for such bounds, Vinogradov was able to use his method with great success in many different problems of number theory.…”

Section: Stochastic Generalization Of the Vinogradov Mean Value Theoremmentioning

confidence: 99%

On Properties of Polynomials in Random Elements

Ulyanov¹

2016

Theory Probab. Appl.

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This paper deals with different properties of polynomials in random elements: bounds for characteristic functionals of polynomials, stochastic generalization of the Vinogradov mean value theorem, the characterization problem, and bounds for probabilities to hit the balls. These results cover the cases when the random elements take values in finite as well as infinite dimensional Hilbert spaces.In section 2 we give the bounds for characteristic functions gn(t, a), which improve (1) so that D increases essentially and becomes proportional to 1/m. It follows from Theorem 4 with two-sided bounds for gn(t, a) that this order of D is the right one. The improvements of (1) in section 2 are proved provided that some conditions are met concerning the type of distribution of X. A desire to remove these conditions leads to the necessity of stochastic generalization of the famous Vinogradov mean value theorem. See the stochastic generalization in section 3. In particular, if X takes values 1, 2, . . . , P , with equal probabilities, our estimate gives the same order with respect to P as Vinogradov's original result. The properties of quadratic forms are considered in sections 4 and 5. They have attracted the attention of researchers in recent years. In section 4 we give sufficient conditions at which a distribution of the finite quadratic form in independent identically distributed symmetric random variables defines a distribution of the basic random variable uniquely. The stability theorem for quadratic forms is proved as well. In section 5 the two-sided bounds are found for the density p(u, a) of |Y − a| 2 , where Y is an H-valued Gaussian random element. The bounds are precise in the sense that the ratio of upper bound to the lower equals eight for all sufficiently large values of u. Thus, the ratio does not depend on the parameters of the distribution of |Y − a| 2 . These bounds imply two-sided bounds for the probabilities P(|Y − a| > r).In this paper we use and discuss Prokhorov's results obtained jointly with various coauthors during 1995-2000 (see [6], [7], [8], [9], [10], [11]). The asymptotic behavior of quadratic and almost quadratic forms that appeared in mathematical statistics is considered in [12]. See [13] as well.

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“…where e(z) = e 2π iz for z ∈ C. Such asymptotic relations can be generalized to multiple Diophantine problems (for more details, see [9,Theorem 1.4]). …”

Section: Introductionmentioning

confidence: 98%

“…where C s,k,a is a positive constant. In particular, this constant can be factored as a product of the local densities associated with the above system defined as in Schmidt [10] and Wooley [12]. When L > 0, for v ∈ R, define λ L (v) = L(1 − L|v|), when |v| L −1 , 0, otherwise.…”

Section: Introductionmentioning

confidence: 99%