Sharp Transitions for Localized Solutions to a Diophantine Inequality

Ataleshvara Bhargava

REVIEW

Sharp Transitions for Localized Solutions to a Diophantine Inequality

Ataleshvara Bhargava

Number Theory · Submitted 18 Aug 2026 · 26 references

Pith's verdict: A genuinely new sharp existence threshold for non-integer-exponent Diophantine inequalities, with a major-arc uniformity gap that is fixable and a few small technical errors.

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arXiv ID2608.18350 (version 1)

Pith ID65KR7ROP, Pith's permanent record of this paper and its review

CitationAtaleshvara Bhargava. "Sharp Transitions for Localized Solutions to a Diophantine Inequality." arXiv:2608.18350 (2026). https://pith.science/paper/65KR7ROP

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Claim paper

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Editor's note

read the letter

Quick take: this is a real result. Bhargava proves that for non-integer θ>2 the almost-diagonal scale Y ≍ X^{1/2} has a sharp constant threshold: c > (⌊s/2⌋(θ−1))^{−1/2} guarantees solutions for all large R, and c below that fails along an infinite sequence. The non-integer setting is genuinely new—Daemen and Wright only did integer Waring—and the proof is a careful Davenport–Heilbronn adaptation with Vinogradov mean value estimates. Theorem 2, the obstruction, is clean and rigorous.

Most of Theorem 1 is in good shape. The minor and trivial arcs are standard but correctly handled; the major-arc approximation and stationary phase are fine. The construction of η± and the IVT argument is clever and does give a non-singular solution. The one place I would want more detail is the step “J0(R,m) ≫ 1” from non-singularity. The paper cites Poulias's truncation lemmas without showing the uniformity conditions. The stress-test worry—that the parameter t on the segment might approach 0 fast enough to make the Jacobian minor vanish in the limit—does not actually occur: v(η−) is bounded above by a negative constant and v(η+) below by a positive constant, so the root is bounded away from both endpoints independently of X, and the alternating coordinates of η+ keep some pair separated by a positive constant. But the paper does not write that out, and a referee should ask for it. As written, it is a genuine gap in exposition, not a fatal flaw.

Two smaller errors. Lemma 4.5 chooses λ_k = α2|θ⋯(θ−k+1)|(X/2)^{θ−k}; for k>θ this goes the wrong way—the negative power makes (X/2)^{θ−k} larger than the actual derivative on the range. Choosing (X+Y)^{θ−k} instead works, with the constant ratio still O(1). And in Lemma 6.5 the binomial expansion has the exponent on (f−f*) as s−i instead of i; the subsequent estimates are unaffected. Neither changes the argument.

The citation pattern is honest and uses standard external tools (Vinogradov, van der Corput, stationary phase, Freeman). No self-citation or fitted-parameter circularity.

Bottom line: with the uniformity argument added and the small errors fixed, the theorem stands. This deserves a serious referee; I would send it out for full peer review rather than desk reject.

Referee report

Significance

If the proof is completed, this is a strong and attractive result: it gives an essentially sharp constant for the Fourier-analytic transition in a fractional-exponent analogue of Waring's problem, improving on the corresponding Daemen–Wright thresholds in the integer-degree setting and clarifying the heuristic failure described in Section 2. The paper makes good use of modern tools — the resolved main conjecture in Vinogradov's mean value theorem, Bourgain–Demeter–Guth, and Freeman's kernel — and the overall strategy is coherent. The main theorem is falsifiable and the constants are explicit in terms of s and θ. The result is likely correct after filling the gaps identified below; the missing major-arc uniformity argument and the minor-arc derivative-test issue are both local and repairable. The paper also offers a thoughtful discussion of the critical case c=ω and of open problems, which adds value.

Major objections (2)

[Section 6, proof that J0(R,m) ≫ 1] The lower bound J0(R,m) ≫ 1 is load-bearing for Theorem 1, but the text only proves the existence of one non-singular interior solution η to u(γ)=v(γ)=0. The Jacobian minor |det A_{i,j}| is comparable to θ(θ−1)|η_i−η_j| + O(X^{−1/2}), and the proof establishes only that some pair of coordinates is distinct. The cited "standard truncation and approximation argument" (Lemmas 3.6.4–3.6.7 of [14]) requires a uniform lower bound on the Jacobian rank and a uniformly positive (s−2)-dimensional volume of the zero set near η; without such bounds, J0(R,m) could in principle be o(1) and the major arc would fail to dominate the error terms. The missing step is, however, available: from v(η)=θm/c² + binom(θ,2)‖η‖₂² + O(X^{−1/2}) and m ≤ −μ one obtains ‖η‖₂ ≥ c₀ > 0, and since u(η)=0 implies the coordinates have mean O(Y^{−1}), the maximal pair distance is ≥ c₀/√s. This would give the required uniform determinant lower bound. Alternatively, one can bound the IVT parameter t away from 0 and 1 using the uniform signs of v(η_−) and v(η_+). The argument should be written out explicitly.
[Lemma 4.5] In the second case of the proof, λ_k is chosen as α2|θ⋯(θ−k+1)|(X/2)^{θ−k}. When k>θ — which occurs for θ∈(2,3) when λ is near ν and k=3 — the function (T+x)^{θ−k} is decreasing in x on [1,2Y], so (X/2)^{θ−k} is an upper bound rather than a lower bound for the derivative size. Consequently the displayed inequality λ_k ≤ |g^{(k)}(x)| ≤ 4^{θ−k}λ_k fails in that range because 4^{θ−k}<1, and Lemma 4.4 cannot be applied as written. This affects the minor-arc bound on which Theorem 1 relies. The issue is repairable, for example by taking λ_k = α2|θ⋯(θ−k+1)|(X+Y)^{θ−k} when θ−k<0, so that the ratio of the upper and lower derivative bounds is 1+O(X^{−1/2}); the repair should be written out and the resulting Y-exponent checked.

Minor objections (5)

[Lemma 6.5] In the binomial expansion of (f* + (f−f*))^s, the displayed term (f−f*)^{s−i} should be (f−f*)^i; the subsequent error estimate uses |f*|^{s−i}, consistent with the corrected exponent.
[Section 6, formula for ‖η_−‖] The displayed identity ‖η_−‖_{ℓ2}² = 4m²/(sY²) should read m²/(sY²), and the corresponding term in the formula for v(η_−) should have m² rather than 4m. The conclusion is unaffected because this term is O(Y^{−2}).
[Lemma 4.6] The statement writes H_±(Y,m;m); by the definition on page 8 the first argument should be R, not Y.
[Section 6, implicit-function-theorem conclusion] The text says there is "an (s−2)-dimensional subspace L of positive (s−2)-volume"; the implicit function theorem yields a submanifold, not a linear subspace, and the wording should be corrected.
[Page 3 and miscellaneous typos] Minor typographical issues include "Theorem 2 turns out the be surprising" (should be "to be") and "satisfies the conclusion one Lemma 3.1" (should be "of Lemma 3.1").

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The paper

The paper claims

A single critical constant separates guaranteed solutions from arbitrarily long gaps in a fractional-power Diophantine inequality.

The reading

This paper studies how many ways a large real number $R$ can be approximated by a sum of $s$ powers $x_i^\theta$ with non-integer $\theta>2$, when every variable is forced into a narrow window $X-Y < x_i \leq X+Y$ around the diagonal $X=(R/s)^{1/\theta}$. With window width $Y=c\sqrt{X}$, it proves a critical constant $\omega=(\lfloor s/2\rfloor(\theta-1))^{-1/2}$: for $c>\omega$ and enough variables, every sufficiently large $R$ has such solutions and the count grows like $X^{1-\theta}Y^{s-1}$; for $c<\omega$, infinitely many $R$ have none. The threshold is sharp except at $c=\omega$, left open. This matters because existence of integer approximations becomes a yes/no question governed by a single constant, and the zero case contradicts the naive heuristic that solutions should appear as soon as the window has enough volume.

What carries the argument

The load-bearing object is the exponential sum $f(\alpha_1,\alpha_2)=\sum_{X-Y<x\leq X+Y} e((x-X)\alpha_1+\alpha_2(x^\theta-X^\theta-\theta(x-X)X^{\theta-1}))$, whose $s$-th power is integrated against a smoothing kernel $K_\pm(\alpha_2)$ that converts the inequality $|\sum x_i^\theta-R|<\tau$ into a count. On the major arcs, $f$ is replaced by an integral approximation $f^*$, and the count is reduced to a two-dimensional integral $J_0(R,m)$ whose positivity is proved by finding a non-singular real solution $(\eta_1,\dots,\eta_s)$ to the system $\sum \gamma_i=m/Y$ and $\sum G(Y\gamma_i)=Z$, where $G(t)=(X+t)^\theta-X^\theta-\theta tX^{\theta-1}$. The sign of the quadratic form $\sum \gamma_i^2$, evaluated at two constructed points, changes exactly when $c$ crosses $\omega$, producing the transition. The minor and trivial arc bounds come from mean value estimates for exponential sums and a derivative test for exponential sums.

Extended reading notes

Core claim

The central claim is Theorems 1 and 2: the inequality family has a sharp transition at $\omega$. If $c>\omega$ and $s \geq (\lfloor 2\theta\rfloor+1)(\lfloor 2\theta\rfloor+2)+1$, then for all sufficiently large $R$ one has $N^\tau_{s,\theta}(R,c\sqrt{X}) \gg_{\theta,s} \tau X^{1-\theta}Y^{s-1}$. If $c<\omega$, a construction with $R=sM^\theta+s\theta M^{\theta-1}$ produces infinitely many $R_n\to\infty$ with $N^\tau_{s,\theta}(R_n,c\sqrt{X_n})=0$. The constant $\omega$ emerges from the balance between a linear and a quadratic term in the Taylor expansion of $(X+y)^\theta$ around $X$; this balance is the same mechanism that makes the major-arc integral change sign.

Load-bearing premise

The proof that the major-arc integral $J_0(R,m)$ is bounded below by a positive constant assumes that a standard truncation argument, taken from Lemmas 3.6.4 through 3.6.7 of [14], applies uniformly in $X$ and that the Jacobian determinant at the constructed solution is bounded away from zero; the paper proves existence and non-singularity but does not explicitly verify that uniform lower bound.

Editorial extensions

If this is right

The optimal window width for guaranteed existence is $Y \asymp \sqrt{X}$ up to a constant factor; any smaller constant in front of $\sqrt{X}$ produces arbitrarily large gaps.
For $c>\omega$, the lower bound $N \gg X^{1-\theta}Y^{s-1}$ has the order of magnitude one expects from the circle method, so the count is not merely positive but grows polynomially.
Theorem 2 requires only $s\geq 2$, so the obstruction for small $c$ is a geometric mismatch, not a shortage of variables.
The equality case $c=\omega$ is not settled; the paper suggests the behavior may depend on an additional power $X^\delta$ beyond $\sqrt{X}$.
The same dichotomy is likely to hold for the integer-power analogue, with the same mechanism but a different critical constant.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

One testable extension: for $c<\omega$, measure the density of the exceptional set of $R$ with no solutions; the construction here only shows the set is infinite, but it may well have positive density.
For intermediate numbers of variables, $\theta < s \leq 2\theta-1$, the exponent $X^{1-\theta}Y^{s-1}$ becomes negative, so if solutions exist the correct width may be $X^\kappa$ with $\kappa>1/2$ depending on $s$; this regime is out of reach of the present method.
Allowing the tolerance $\tau$ to grow with $R$ could reduce the required number of variables or the window constant, mirroring what happens in shifted sum-of-powers problems.
The sharpness at $\omega$ suggests a genuine phase transition; numerical searches near $c=\omega$ for specific $\theta$ and $s$ could reveal whether the transition is abrupt or accompanied by sparse sporadic solutions.

Original abstract

For fixed $\tau > 0$, non-integer $\theta > 2$ and large enough $s$, we investigate the number of solutions to the Diophantine inequality $|x_1^{\theta}+\cdots +x_s^{\theta} - R| < \tau$ as $R \to \infty$. Here, we restrict the variables $x_i$ in the ``almost diagonal" range $X-Y < x_i \leq X+Y$ for $i = 1, \ldots, s$, where $X = (R/s)^{1/\theta}$ and $Y \asymp \sqrt{X}$. Let $\omega = (\lfloor s/2 \rfloor(\theta-1))^{-1/2}$. We will show that if $Y = c\sqrt{X}$ for some $c > \omega$ then for sufficiently large $R$ there must always exist solutions, but if $c < \omega$ then there exist arbitrarily large positive $R$ for which there are no solutions. Our result is thus essentially sharp, with the exception of $c = \omega$. This work is analogous to the results of Daemen and Wright in which similar statements are proved for Waring's problem, though our results are likely somewhat stronger than what is possible in that setting. We discuss other related results and further work to be done.

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The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; show every role.

For fixed $\tau > 0$, non-integer $\theta > 2$ and large enough $s$, we investigate the number of solutions to the Diophantine inequality $|x_1^{\theta}+\cdots +x_s^{\theta} - R| < \tau$ as $R \to \infty$. Here, we restrict the variables $x_i$ in the ``almost diagonal" range $X-Y < x_i \leq X+Y$ for $i = 1, \ldots, s$, where $X = (R/s)^{1/\theta}$ and $Y \asymp \sqrt{X}$. Let $\omega = (\lfloor s/2 \rfloor(\theta-1))^{-1/2}$. We will show that if $Y = c\sqrt{X}$ for some $c > \omega$ then for sufficiently large $R$ there must always exist solutions, but if $c < \omega$ then there exist arbitrarily large positive $R$ for which there are no solutions. Our result is thus essentially sharp, with the exception of $c = \omega$. This work is analogous to the results of Daemen and Wright in which similar statements are proved for Waring's problem, though our results are likely somewhat stronger than what is possible in that setting. We discuss other related results and further work to be done.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
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s0	For fixed $\tau > 0$, non-integer $\theta > 2$ and large enough $s$, we investigate the number of solutions to the Diophantine inequality $\|x_1^{\theta}+\cdots +x_s^{\theta} - R\| < \tau$ as $R \to \infty$.	contribution 0.67	0.73	0.03	unclear 0.56
s1	Here, we restrict the variables $x_i$ in the ``almost diagonal" range $X-Y < x_i \leq X+Y$ for $i = 1, \ldots, s$, where $X = (R/s)^{1/\theta}$ and $Y \asymp \sqrt{X}$.	method 0.73	0.76	0.02	asserted 0.92
s2	Let $\omega = (\lfloor s/2 \rfloor(\theta-1))^{-1/2}$.	method 0.55	0.48	0.02	asserted 0.58
s3	We will show that if $Y = c\sqrt{X}$ for some $c > \omega$ then for sufficiently large $R$ there must always exist solutions, but if $c < \omega$ then there exist arbitrarily large positive $R$ for which there are no solutions.	result 1.00	0.93	0.93	asserted 1.00
s4	Our result is thus essentially sharp, with the exception of $c = \omega$.	interpretation 0.75	0.89	0.27	asserted 0.96
s5	This work is analogous to the results of Daemen and Wright in which similar statements are proved for Waring's problem, though our results are likely somewhat stronger than what is possible in that setting.	background 0.65	0.17	0.03	hedged possibility 0.88
s6	We discuss other related results and further work to be done.	future work 1.00	0.19	0.01	unclear 0.58

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 1.00.

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arXiv listing abstract v1

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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

References

Works cited (26)

[14]

K. Poulias. On Diophantine problems involving fractional powers of int egers. Phd thesis, Uni- versity of Bristol, 2021. Available at https://research-information.bris.ac.uk/en/studentTheses/ on-diophantine-problems-involving-fractional-powers -of-integers/

View on Pith 2021
[1]

T. M. Apostol. Mathematical analysis . Addison-Wesley Publishing Co., Reading, Mass.-London-D on Mills, Ont., second edition, 1974

View on Pith 1974
[2]

G. I. Arkhipov and A. N. Zhitkov. Waring’s problem with no nintegral exponent. Izv. Akad. Nauk SSSR Ser. Mat., 48(6):1138–1150, 1984

View on Pith 1984
[3]

K. D. Biggs. Almost equal summands in Waring’s problem wi th shifts. Monatsh. Math. , 188(1):31–35, 2019

View on Pith 2019
[4]

J. Bourgain, C. Demeter, and L. Guth. Proof of the main con jecture in Vinogradov’s mean value theorem for degrees higher than three. Ann. of Math. (2) , 184(2):633–682, 2016

View on Pith 2016
[5]

D. Daemen. The asymptotic formula for localized solutio ns in Waring’s problem and approximations to Weyl sums. Bull. Lond. Math. Soc. , 42(1):75–82, 2010

View on Pith 2010
[6]

D. Daemen. Localized solutions in Waring’s problem: the lower bound. Acta Arith., 142(2):129–143, 2010

View on Pith 2010
[7]

H. Davenport. Analytic methods for Diophantine equations and Diophantin e inequalities . Cambridge Math- ematical Library. Cambridge University Press, Cambridge, second edition, 2005. With a foreword by R. C. Vaughan, D. R. Heath-Brown and D. E. Freeman, Edited and prep ared for publication by T. D. Browning

View on Pith 2005
[8]

J.-M. Deshouillers. Probl` eme de Waring avec exposants non entiers. Bull. Soc. Math. France , 101:285–295, 1973

View on Pith 1973
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D. E. Freeman. Asymptotic lower bounds for Diophantine i nequalities. Mathematika, 47(1-2):127–159, 2000

View on Pith 2000
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D. E. Freeman. Asymptotic lower bounds and formulas for Diophantine inequalities. In Number theory for the millennium, II (Urbana, IL, 2000) , pages 57–74. A K Peters, Natick, MA, 2002

View on Pith 2000
[11]

S. W. Graham and G. Kolesnik. van der Corput’s method of exponential sums , volume 126 of London Mathe- matical Society Lecture Note Series . Cambridge University Press, Cambridge, 1991

View on Pith 1991
[12]

T.S. K¨ ufner.A real eﬀective version of the Fre ˘iman-Scourﬁeld Theorem. Phd thesis, Georg-August University School of Science, G¨ ottingen, 2025. Available at http://dx.doi.org/10.53846/goediss-11614

View on Pith doi:10.53846/goediss-11614 2025
[13]

C. Poulias. Diophantine inequalities of fractional de gree. Mathematika, 67(4):949–980, 2021. SHARP TRANSITIONS FOR LOCALIZED SOLUTIONS TO A DIOPHANTINE INEQUALITY 23

View on Pith 2021
[15]

O. Robert. On van der Corput’s k-th derivative test for exponential sums. Indag. Math. (N.S.) , 27(2):559–589, 2016

View on Pith 2016
[16]

J. Salmensuu. On the Waring-Goldbach problem with almo st equal summands. Mathematika, 66(2):255–296, 2020

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E. M. Stein. Harmonic analysis: real-variable methods, orthogonality , and oscillatory integrals , volume 43 of Princeton Mathematical Series . Princeton University Press, Princeton, NJ, 1993. With the assistance of Timothy S. Murphy, Monographs in Harmonic Analysis, III

View on Pith 1993
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J. G. Van der Corput. Zur Methode der station¨ aren Phase . Erste Mitteilung Einfache Integrale. Compositio Math., 1:15–38, 1935

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R. C. Vaughan. The Hardy-Littlewood method , volume 125 of Cambridge Tracts in Mathematics . Cambridge University Press, Cambridge, second edition, 1997

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[20]

I. M. Vinogradov. The method of trigonometrical sums in the theory of numbers . Dover Publications, Inc., Mineola, NY, 2004. Translated from the Russian, revised and annotated by K. F. Roth and Anne Davenport, Reprint of the 1954 translation

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N. Watt. Exponential sums and the Riemann zeta-functio n. II. J. London Math. Soc. (2) , 39(3):385–404, 1989

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[22]

B. Wei and T. D. Wooley. On sums of powers of almost equal p rimes. Proc. Lond. Math. Soc. (3) , 111(5):1130– 1162, 2015

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T. D. Wooley. On Diophantine inequalities: Freeman’s a symptotic formulae. In Proceedings of the Session in Analytic Number Theory and Diophantine Equations , volume 360 of Bonner Math. Schriften , pages 1–32. Univ. Bonn, Bonn, 2003

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T. D. Wooley. Nested eﬃcient congruencing and relative s of Vinogradov’s mean value theorem. Proc. Lond. Math. Soc. (3) , 118(4):942–1016, 2019

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E. M. Wright. Proportionality conditions in Waring’s p roblem. Math. Z. , 38(1):730–746, 1934

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E. M. Wright. The representation of a number as a sum of fo ur ‘almost equal’ squares. Quarterly Journal of Mathematics, 8:278–279, 1937. Department of Mathematics, Purdue University, West Lafaye tte, IN, USA. Email address : bharga37@purdue.edu

View on Pith 1937

Discussion

Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.

[1] [14]

K. Poulias. On Diophantine problems involving fractional powers of int egers. Phd thesis, Uni- versity of Bristol, 2021. Available at https://research-information.bris.ac.uk/en/studentTheses/ on-diophantine-problems-involving-fractional-powers -of-integers/

View on Pith 2021

[2] [1]

T. M. Apostol. Mathematical analysis . Addison-Wesley Publishing Co., Reading, Mass.-London-D on Mills, Ont., second edition, 1974

View on Pith 1974

[3] [2]

G. I. Arkhipov and A. N. Zhitkov. Waring’s problem with no nintegral exponent. Izv. Akad. Nauk SSSR Ser. Mat., 48(6):1138–1150, 1984

View on Pith 1984

[4] [3]

K. D. Biggs. Almost equal summands in Waring’s problem wi th shifts. Monatsh. Math. , 188(1):31–35, 2019

View on Pith 2019

[5] [4]

J. Bourgain, C. Demeter, and L. Guth. Proof of the main con jecture in Vinogradov’s mean value theorem for degrees higher than three. Ann. of Math. (2) , 184(2):633–682, 2016

View on Pith 2016

[6] [5]

D. Daemen. The asymptotic formula for localized solutio ns in Waring’s problem and approximations to Weyl sums. Bull. Lond. Math. Soc. , 42(1):75–82, 2010

View on Pith 2010

[7] [6]

D. Daemen. Localized solutions in Waring’s problem: the lower bound. Acta Arith., 142(2):129–143, 2010

View on Pith 2010

[8] [7]

H. Davenport. Analytic methods for Diophantine equations and Diophantin e inequalities . Cambridge Math- ematical Library. Cambridge University Press, Cambridge, second edition, 2005. With a foreword by R. C. Vaughan, D. R. Heath-Brown and D. E. Freeman, Edited and prep ared for publication by T. D. Browning

View on Pith 2005

[9] [8]

J.-M. Deshouillers. Probl` eme de Waring avec exposants non entiers. Bull. Soc. Math. France , 101:285–295, 1973

View on Pith 1973

[10] [9]

D. E. Freeman. Asymptotic lower bounds for Diophantine i nequalities. Mathematika, 47(1-2):127–159, 2000

View on Pith 2000

[11] [10]

D. E. Freeman. Asymptotic lower bounds and formulas for Diophantine inequalities. In Number theory for the millennium, II (Urbana, IL, 2000) , pages 57–74. A K Peters, Natick, MA, 2002

View on Pith 2000

[12] [11]

S. W. Graham and G. Kolesnik. van der Corput’s method of exponential sums , volume 126 of London Mathe- matical Society Lecture Note Series . Cambridge University Press, Cambridge, 1991

View on Pith 1991

[13] [12]

T.S. K¨ ufner.A real eﬀective version of the Fre ˘iman-Scourﬁeld Theorem. Phd thesis, Georg-August University School of Science, G¨ ottingen, 2025. Available at http://dx.doi.org/10.53846/goediss-11614

View on Pith doi:10.53846/goediss-11614 2025

[14] [13]

C. Poulias. Diophantine inequalities of fractional de gree. Mathematika, 67(4):949–980, 2021. SHARP TRANSITIONS FOR LOCALIZED SOLUTIONS TO A DIOPHANTINE INEQUALITY 23

View on Pith 2021

[15] [15]

O. Robert. On van der Corput’s k-th derivative test for exponential sums. Indag. Math. (N.S.) , 27(2):559–589, 2016

View on Pith 2016

[16] [16]

J. Salmensuu. On the Waring-Goldbach problem with almo st equal summands. Mathematika, 66(2):255–296, 2020

View on Pith 2020

[17] [17]

E. M. Stein. Harmonic analysis: real-variable methods, orthogonality , and oscillatory integrals , volume 43 of Princeton Mathematical Series . Princeton University Press, Princeton, NJ, 1993. With the assistance of Timothy S. Murphy, Monographs in Harmonic Analysis, III

View on Pith 1993

[18] [18]

J. G. Van der Corput. Zur Methode der station¨ aren Phase . Erste Mitteilung Einfache Integrale. Compositio Math., 1:15–38, 1935

View on Pith 1935

[19] [19]

R. C. Vaughan. The Hardy-Littlewood method , volume 125 of Cambridge Tracts in Mathematics . Cambridge University Press, Cambridge, second edition, 1997

View on Pith 1997

[20] [20]

I. M. Vinogradov. The method of trigonometrical sums in the theory of numbers . Dover Publications, Inc., Mineola, NY, 2004. Translated from the Russian, revised and annotated by K. F. Roth and Anne Davenport, Reprint of the 1954 translation

View on Pith 2004

[21] [21]

N. Watt. Exponential sums and the Riemann zeta-functio n. II. J. London Math. Soc. (2) , 39(3):385–404, 1989

View on Pith 1989

[22] [22]

B. Wei and T. D. Wooley. On sums of powers of almost equal p rimes. Proc. Lond. Math. Soc. (3) , 111(5):1130– 1162, 2015

View on Pith 2015

[23] [23]

T. D. Wooley. On Diophantine inequalities: Freeman’s a symptotic formulae. In Proceedings of the Session in Analytic Number Theory and Diophantine Equations , volume 360 of Bonner Math. Schriften , pages 1–32. Univ. Bonn, Bonn, 2003

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