Discrete Fourier restriction via Efficient Congruencing

Wooley, Trevor D.

doi:10.1093/imrn/rnw031

Cited by 20 publications

(36 citation statements)

References 28 publications

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“…The contribution of the minor arcs

frakturm

is easily estimated. Thus, as in [, equation (7.1)] (based on [, § 9]), we find that for each

ε > 0

one has

\underset{bold-italicα \in m}{trueprefixsup} false| f (bold-italicα; X) false| ≪ X^{1 - τ + ε},

where

τ^{- 1} = 8 k^{2}

. In this way, when

2 s > k (k + 1)

, one finds that

\begin{matrix} true \\ right \int_{frakturm} {| f false(α; X false) |}^{2 s} normald bold-italicα & left ≪ {(sup_{α \in frakturm} | f false(α; X false) |)}^{2 s - k false(k + 1 false)} \oint {false| f (bold-italicα; X) false|}^{k false(k + 1 false)} d α \\ left ≪ X^{2 s - k false(k + 1 false) / 2 + 2 ε - false(2 s - k false(k + 1 false) false) τ} . \end{matrix}

Thus

0true \int_{m} {false| f (bold-italicα; X) false|}^{2 s} d α

…”

Section: Mean Values Of Exponential Sumssupporting

confidence: 61%

“…Theorem also delivers the expected Strichartz inequality established for

s ⩾ k (k + 1)

in , and subsequently in full in . Corollary Suppose that

k \in N

, that

s

is a positive number, and

{false(a_{n} false)}_{n \in double-struckZ}

is a complex sequence.…”

Section: Introductionmentioning

confidence: 63%

“…Adopting standard notation, we write J s,k (X) for the number of integral solutions of the system (1.6) with 1 x i , y i X (1 i s). More generally, when s is not necessarily an integer, we put J s,k (X) = [0,1) k 1 x X Theorem 1.1 also delivers the expected Strichartz inequality established for s k(k + 1) in [61], and subsequently in full in [9]. Corollary 1.4.…”

Section: Introductionmentioning

confidence: 68%

“…In this way, when

2 s > k (k + 1)

, one finds that

\begin{matrix} true \\ right \int_{frakturm} {| f false(α; X false) |}^{2 s} normald bold-italicα & left ≪ {(sup_{α \in frakturm} | f false(α; X false) |)}^{2 s - k false(k + 1 false)} \oint {false| f (bold-italicα; X) false|}^{k false(k + 1 false)} d α \\ left ≪ X^{2 s - k false(k + 1 false) / 2 + 2 ε - false(2 s - k false(k + 1 false) false) τ} . \end{matrix}

Thus

0true \int_{m} {false| f (bold-italicα; X) false|}^{2 s} d α = o (X^{2 s - k (k + 1) / 2}),

provided that we take

ε

sufficiently small in terms of

s

and

k

. On the other hand, as in the proof of [, Lemma 7.1], one finds that when

bold-italicα \in M (q, a) \subseteq M

, one has

f ...

…”

Section: Mean Values Of Exponential Sumsmentioning

confidence: 93%

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

Wooley

2018

Proc. London Math. Soc.

Self Cite

125

141

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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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“…The contribution of the minor arcs

frakturm

is easily estimated. Thus, as in [, equation (7.1)] (based on [, § 9]), we find that for each

ε > 0

one has

\underset{bold-italicα \in m}{trueprefixsup} false| f (bold-italicα; X) false| ≪ X^{1 - τ + ε},

where

τ^{- 1} = 8 k^{2}

. In this way, when

2 s > k (k + 1)

, one finds that

\begin{matrix} true \\ right \int_{frakturm} {| f false(α; X false) |}^{2 s} normald bold-italicα & left ≪ {(sup_{α \in frakturm} | f false(α; X false) |)}^{2 s - k false(k + 1 false)} \oint {false| f (bold-italicα; X) false|}^{k false(k + 1 false)} d α \\ left ≪ X^{2 s - k false(k + 1 false) / 2 + 2 ε - false(2 s - k false(k + 1 false) false) τ} . \end{matrix}

Thus

0true \int_{m} {false| f (bold-italicα; X) false|}^{2 s} d α

…”

Section: Mean Values Of Exponential Sumssupporting

confidence: 61%

“…Theorem also delivers the expected Strichartz inequality established for

s ⩾ k (k + 1)

in , and subsequently in full in . Corollary Suppose that

k \in N

, that

s

is a positive number, and

{false(a_{n} false)}_{n \in double-struckZ}

is a complex sequence.…”

Section: Introductionmentioning

confidence: 63%

Section: Introductionmentioning

confidence: 68%

“…In this way, when

2 s > k (k + 1)

, one finds that

\begin{matrix} true \\ right \int_{frakturm} {| f false(α; X false) |}^{2 s} normald bold-italicα & left ≪ {(sup_{α \in frakturm} | f false(α; X false) |)}^{2 s - k false(k + 1 false)} \oint {false| f (bold-italicα; X) false|}^{k false(k + 1 false)} d α \\ left ≪ X^{2 s - k false(k + 1 false) / 2 + 2 ε - false(2 s - k false(k + 1 false) false) τ} . \end{matrix}

Thus

0true \int_{m} {false| f (bold-italicα; X) false|}^{2 s} d α = o (X^{2 s - k (k + 1) / 2}),

provided that we take

ε

sufficiently small in terms of

s

and

k

. On the other hand, as in the proof of [, Lemma 7.1], one finds that when

bold-italicα \in M (q, a) \subseteq M

, one has

f ...

…”

Section: Mean Values Of Exponential Sumsmentioning

confidence: 93%

See 2 more Smart Citations

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

Wooley

2018

Proc. London Math. Soc.

Self Cite

125

141

View full text Add to dashboard Cite

show abstract

“…Furthermore they have been used by Halász and Turán [4] to get zero-density estimates for the Riemann-Zeta function. Other applications of Vinogradov's mean value theorem include estimates for short mixed character sums, such as found in work of Heath-Brown and Pierce [5] and Kerr [10], as well as contributions to restriction theory worked out by Wooley [20] and Bourgain, Demeter and Guth [2]. In all of these applications it is desirable to have an effective version of Vinogradov's mean value theorem.…”

Section: Introductionmentioning

confidence: 99%

Effective Vinogradov's mean value theorem via efficient boxing

Steiner

2019

Journal of Number Theory

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We combine Wooley's efficient congruencing method with earlier work of Vinogradov and Hua to get effective bounds on Vinogradov's mean value theorem.with 0 < x, y ≤ X. By orthogonality the number of solutions is equal tofor real N, M with M ≥ 1 and ϑ(x) = (x, x 2 , . . . , x k ). Lower bounds for J s,k (X) are well-known and easily proved (see for example [14]). They admit the formIn a recent breakthrough Bourgain, Demeter and Guth [2] have shown that (1.2) is sharp up to a factor X ǫ ; i.e. they have proven the inequality (1.3) J s,k (X) ≪ s,k,ǫ max{X s+ǫ , X 2s− 1 2 k(k+1)+ǫ } 2010 Mathematics Subject Classification. 11P55 (11L07, 11L15, 11D45). Key words and phrases. Exponential sums, Hardy-Littlewood method, Effective Vinogradov's mean value theorem. 1    be an integer and set θ = k −(D+1) . Then we have J s,k (X) ≤C · 2Iterating this theorem combined with the Hardy-Littlewood method one may conclude the following result, which is a special case of Theorem 7.3.Theorem 1.2. Let k ≥ 3, s ≥ 5 2 k 2 + k. Furthermore let X ≥ s 10 . Then we have the estimate J s,k (X) ≤ CX 2s− 1 2 k(k+1) ,

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Restriction estimates of $$\varepsilon $$ ε -removal type for k-th powers and paraboloids

Henriot

Hughes

2018

Math. Ann.

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We obtain restriction estimates of -removal type for the set of k -th powers of integers, and for discrete d -dimensional surfaces of the form which we term ‘ k -paraboloids’. For these surfaces, we obtain a satisfying range of exponents for large values of d , k . We also obtain estimates of -removal type in the full supercritical range for k -th powers and for k -paraboloids of dimension . We rely on a variety of techniques in discrete harmonic analysis originating in Bourgain’s works on the restriction theory of the squares and the discrete parabola.

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Discrete Fourier restriction via Efficient Congruencing

Cited by 20 publications

References 28 publications

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

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

Effective Vinogradov's mean value theorem via efficient boxing

Restriction estimates of $$\varepsilon $$ ε -removal type for k-th powers and paraboloids

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