Roth–Waring–Goldbach

Chow, Samuel Khai Ho

doi:10.1093/imrn/rnw307

Cited by 14 publications

(19 citation statements)

References 32 publications

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“…Proof We mostly follow ideas presented in [, §4]. Recalling and we see that

\begin{matrix} true \\ right V_{q} (a d^{k}, b, d, 0) & = & left \sum_{\begin{matrix} z \in [W] \\ {(z d)}^{k} \equiv b false(\mod W false) \end{matrix}} V_{q}^{'} (a d^{k}, z, 0) \\ = & left \sum_{\begin{matrix} z \in [W] \\ {(z d)}^{k} \equiv b false(\mod W false) \end{matrix}} e_{W q} (a d^{k} z^{k}) S_{q} (a d^{k}, z, 0) \\ = & left \sum_{\begin{matrix} z \in [W] \\ {(z d)}^{k} \equiv b false(\mod W false) \end{matrix}} e_{W q} (a d^{k} z^{k}) S_{q_{1}} (a d^{k} \bar{q_{2}}, z, 0) S_{q_{2}} (a d^{k} \bar{q_{1}}, z, 0) . \end{matrix}

Let

a^{'} = a d^{k} \bar{q_{2}}

h = (q_{1}, W)

q_{1}

…”

Section: Pseudorandomness Conditionmentioning

confidence: 95%

“…As stated in §2 to prove pseudorandomness we split the interval [0, 1] into minor and major arcs and treat those sets differently. For the minor arcs, we use an application of [, Lemma 1] and for the major arcs, we develop some ideas that are from [, §4; , §4]. Lemma Let

s ⩾ k^{2} + k + 1

and let

f_{b} : false[N false] \to R

be as in .…”

Section: Proof Of the Main Theoremmentioning

confidence: 99%

“…To establish Lemma we will use a strategy which is very similar to what Sam Chow uses in his case [, §5]. Most significant differences are that we have our function defined on short interval and that we need to use Bourgain's strategy (see [, §4]) only once, since we have power saving on the minor arcs.…”

Section: Restriction Estimatementioning

confidence: 99%

“…To obtain Lemma from Lemma we will use a strategy that was originally introduced by Bourgain [, §4] and is used similarly to our case in [, §6; , §5]. The following lemma is our version of Bourgain's argument.…”

Section: Restriction Estimatementioning

confidence: 99%

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On the Waring–goldbach Problem With Almost Equal Summands

Salmensuu

2020

Mathematika

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“…Proof We mostly follow ideas presented in [, §4]. Recalling and we see that

\begin{matrix} true \\ right V_{q} (a d^{k}, b, d, 0) & = & left \sum_{\begin{matrix} z \in [W] \\ {(z d)}^{k} \equiv b false(\mod W false) \end{matrix}} V_{q}^{'} (a d^{k}, z, 0) \\ = & left \sum_{\begin{matrix} z \in [W] \\ {(z d)}^{k} \equiv b false(\mod W false) \end{matrix}} e_{W q} (a d^{k} z^{k}) S_{q} (a d^{k}, z, 0) \\ = & left \sum_{\begin{matrix} z \in [W] \\ {(z d)}^{k} \equiv b false(\mod W false) \end{matrix}} e_{W q} (a d^{k} z^{k}) S_{q_{1}} (a d^{k} \bar{q_{2}}, z, 0) S_{q_{2}} (a d^{k} \bar{q_{1}}, z, 0) . \end{matrix}

Let

a^{'} = a d^{k} \bar{q_{2}}

h = (q_{1}, W)

q_{1}

…”

Section: Pseudorandomness Conditionmentioning

confidence: 95%

s ⩾ k^{2} + k + 1

and let

f_{b} : false[N false] \to R

be as in .…”

Section: Proof Of the Main Theoremmentioning

confidence: 99%

Section: Restriction Estimatementioning

confidence: 99%

Section: Restriction Estimatementioning

confidence: 99%

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On the Waring–goldbach Problem With Almost Equal Summands

Salmensuu

2020

Mathematika

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“…We refer the reader to [16,17] for recent progress on the Waring-Goldbach problem due to Kumchev and Wooley. There is also [5] by Chow regarding prime solutions of certain diagonal equations by a transference principle approach. For the case of quadratic forms, there is a result due to Zhao [25].…”

Section: Introductionmentioning

confidence: 99%

Prime Solutions to Polynomial Equations in Many Variables and Differing Degrees

Yamagishi

2018

Forum of Mathematics, Sigma

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Let $\mathbf{f}=(f_{1},\ldots ,f_{R})$ be a system of polynomials with integer coefficients in which the degrees need not all be the same. We provide sufficient conditions for which the system of equations $f_{j}(x_{1},\ldots ,x_{n})=0~(1\leqslant j\leqslant R)$ satisfies a general local to global type statement, and has a solution where each coordinate is prime. In fact we obtain the asymptotic formula for number of such solutions, counted with a logarithmic weight, under these conditions. We prove the statement via the Hardy–Littlewood circle method. This is a generalization of the work of Cook and Magyar [‘Diophantine equations in the primes’, Invent. Math.198 (2014), 701–737], where they obtained the result when the polynomials of $\mathbf{f}$ all have the same degree. Hitherto, results of this type for systems of polynomial equations involving different degrees have been restricted to the diagonal case.

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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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Roth–Waring–Goldbach

Abstract: We use Green's transference principle to show that any subset of the dth powers of primes with positive relative density contains nontrivial solutions to a translation-invariant linear equation in d 2 + 1 or more variables, with explicit quantitative bounds.

Cited by 14 publications

References 32 publications

On the Waring–goldbach Problem With Almost Equal Summands

On the Waring–goldbach Problem With Almost Equal Summands

Prime Solutions to Polynomial Equations in Many Variables and Differing Degrees

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

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