Improving the tenth-moment bound $N_{10}(\delta,\Delta)$ for two-scale exponential sums: reducing the $(\delta+\Delta)N^6$ term in Bourgain's inequality (4.18) and closing the Theorem 9 range gap between $N^{-33/18}$ and $N^{-11/7}$

This problem addresses the open core of the acknowledged limitations in Jean Bourgain's paper 'Decoupling inequalities and some mean-value theorems' (Journal d'Analyse Mathématique 133 (2017), 313-334). For exponential sums with phases $kx_0+k^2x_1+\delta^{-1}\phi_2(k/N)x_2+\Delta^{-1}\phi_3(k/N)x_3$ carrying two frequency scales $\delta<\Delta$, the paper proves the tenth-moment bound $N_{10}(\delta,\Delta)\ll[\delta\Delta^{3/4}N^7+(\delta+\Delta)N^6+N^5]N^{\varepsilon}$ under the Wronskian nondegeneracy condition $W(\phi_2''',\phi_3''')\neq0$, and shows $N_{10}(\delta,\delta N)\ll N^{5+\varepsilon}$ for $N^{-33/18}\ge\delta\ge N^{-2}$, while a Diophantine lower bound forces failure for $\delta>N^{-11/7}$. The author states that the presentation, in particular the leading $(\delta+\Delta)N^6$ term, can likely be improved. Later work (Demeter-Guth-Wang 2020; Demeter 2022; Jung 2023; Glidewell 2026) treats these tenth-moment problems as open: the paper's approach is characterized there as ad hoc, restricted to specific exponents, and suboptimal, and the tools developed since (small cap decoupling, sharp $L^{10}$ decoupling for the twisted cubic) do not cover its non-moment-curve phases. Neither the improvement of (4.18) nor the determination of the true $\delta$-range of the $N^{5+\varepsilon}$ bound has been achieved.

Background

This question comes from the interaction of two areas: decoupling inequalities in harmonic analysis and the first spacing problem in the analytic theory of the Riemann zeta function.

Exponential sums and their mean values. Write $e(t)=e^{2\pi i t}$ and let $k\sim N$ denote that the integer $k$ ranges over a dyadic block $[N,2N)$. Sharp bounds for averaged powers of exponential sums $\sum_{k\sim N}e(P(k))$, with $P$ a polynomial- or monomial-like phase, are a central object of analytic number theory. Bourgain's paper studies two-scale mean values: with two frequency parameters $\delta<\Delta$ and fixed functions $\phi_2,\phi_3$ on $[0,1]$, let $N_{10}(\delta,\Delta)$ be the tenth moment, over the unit box in $(x_0,x_1,x_2,x_3)$, of the exponential sum with phase $kx_0+k^2x_1+\frac{1}{\delta}\phi_2(\frac{k}{N})x_2+\frac{1}{\Delta}\phi_3(\frac{k}{N})x_3$. (The paper's own symbol $N_{10}(\delta,\Delta)$, defined at its (0.2) and used in its Theorems 3, 4 and 9, denotes the special case $\phi_2(t)=t^{3/2}$, $\phi_3(t)=t^{1/2}$; Theorem 4 states its bound for general phases satisfying the Wronskian condition below.) The case of interest for applications is $\phi_2(t)=t^{3/2}$, $\phi_3(t)=t^{1/2}$, $\Delta=N\delta$, i.e. the curve $(t,t^2,t^{3/2},t^{1/2})$:

$$N_{10}(\delta,N\delta)=\int_{[0,1]^4}\Big|\sum_{n\sim N}e\Big(x_0n+x_1n^2+x_2\tfrac{1}{\delta}\big(\tfrac{n}{N}\big)^{3/2}+x_3\tfrac{1}{N\delta}\big(\tfrac{n}{N}\big)^{1/2}\Big)\Big|^{10}\,dx_0\,dx_1\,dx_2\,dx_3.$$

Why these sums: the first spacing problem. The Bombieri-Iwaniec method for bounding the Riemann zeta function $\zeta$ on the critical line $\Re s=\frac{1}{2}$ reduces the problem to combinatorial-analytic questions about the spacing of points on curves; the first of these, the first spacing problem, is captured by mean values of exactly the above shape, and the zeta application draws on the range $\delta\sim N^{-2}$. The paper also improves the two-variable moments $I_p(\lambda)=\iint\big|\sum_{n\sim N}e(n^2x+\lambda n^4y)\big|^p\,dx\,dy$ introduced in earlier work of Robert and Sargos on Weyl sums (the power sums $\sum_{n\le N}e(\alpha n^k)$; cited as [R-S] there, where $I_{10}\ll N^{49/8+\varepsilon}$ is proven): Theorem 10 gives $I_{10}\ll N^{17/3+\varepsilon}$ (the case $\lambda=1$), Theorem 11 gives $I_8\le I_8(N^{-7/3})\ll N^{13/3+\varepsilon}$, and the final section also establishes $I_{10}\le I_{10}(N^{-5/3})\ll N^{17/3+\varepsilon}$ (the introduction's (0.18)); these sharpen the exponents entering Weyl-type inequalities (upper bounds for power Weyl sums in terms of rational approximations to the frequency) for eighth and tenth powers. Decoupling. An $\ell^2(L^p)$ decoupling inequality bounds the $L^p$ norm of a superposition of wave packets with rapidly oscillating phases supported near a curved surface by a constant times the $\ell^2$ combination of the pieces' $L^p$ norms; since the solution of the main conjecture in Vinogradov's mean value theorem (the classical mean value problem for $\sum_{k\le N}e(kx_1+\dots+k^nx_n)$), decoupling has been the standard route to moment bounds for exponential sums. Bourgain's paper applies an $L^{10/3}$ decoupling for hypersurfaces in $\mathbb{R}^4$ at the level of tenth moments, exploiting the factorization $10=\frac{10}{3}\times3$; later authors (Jung) describe this as a soft decoupling argument, i.e. one that loses sharpness. What the paper proves. Assume the Wronskian nondegeneracy condition $W(\phi_2''',\phi_3''')=\phi_2'''\phi_3''''-\phi_2''''\phi_3'''\neq0$ on $[0,1]$ and $\delta<\Delta$. Theorem 4 of the paper gives (4.18):

$$N_{10}(\delta,\Delta)\ll\big[\delta\Delta^{3/4}N^7+(\delta+\Delta)N^6+N^5\big]N^{\varepsilon}.$$

In the applications $\Delta$ is much larger than $\delta$, so the middle term $(\delta+\Delta)N^6$ dominates; the paper develops an additional argument to reduce its weight and states, immediately after Theorem 4, 'It is likely that our presentation can be improved at this point.' A companion result, Theorem 9, shows $N_{10}(\delta,\delta N)\ll N^{5+\varepsilon}$ throughout $N^{-2}\le\delta\le N^{-33/18}$ (numerically $11/7\approx1.571<33/18\approx1.833$). But the Remark following Theorem 4 derives, from an explicit Diophantine system on short intervals, the lower bound $N_{10}(\delta,\Delta)\gtrsim\delta\Delta^{3/4}N^7$, which at $\Delta=\delta N$ exceeds $N^5$ precisely when $\delta>N^{-11/7}$; hence the $N^{5+\varepsilon}$ conclusion must fail there. Between the exponents $33/18$ and $11/7$ the truth is unknown: the middle term $(\delta+\Delta)N^6$ of (4.18), which at $\Delta=\delta N$ equals about $\delta N^7$ and first exceeds $N^5$ as soon as $\delta$ moves above $N^{-2}$, is exactly what blocks the range extension, which is why the two components of the problem below are one scientific objective rather than two unrelated tasks. Later developments. The small cap decoupling theory of Demeter-Guth-Wang (2020) — decoupling for caps smaller than the canonical curvature-induced ones, with applications to moments of periodic exponential sums over subdomains smaller than the full period — generalizes these questions (their Conjecture 2.5 for the moment curve $t\mapsto(t,t^2,\dots,t^n)$ on spatially truncated boxes); they describe the small-cap arguments of Bourgain's paper as 'ad hoc ... restricted to specific exponents, and ... ultimately reduced to canonical scale decouplings for higher dimensional manifolds', and note that for the case of dimension $n=3$ with truncation exponent $\beta=2$ only two arguments were known, one of Bombieri-Iwaniec and one in Bourgain's paper. Demeter (2022) formulates a square-root-cancellation conjecture (square-root cancellation: the expected regime in which a sum of $N$ unit-modulus terms typically has size about $N^{1/2}$, so that $p$-th moments over full periods are about $N^{p/2}$) for nondegenerate curves in $\mathbb{R}^4$ only in the range $11\le p\le12$, proves that $p\ge11$ is necessary when the spatial box is truncated — where constructive interference (many summands aligning in phase to produce values much larger than $N^{1/2}$) forces larger moments — and explicitly flags the $p=10$ moments as an open target, citing Bourgain's paper as the recent progress. Jung (2023) proves the sharp $\ell^{10}(L^{10})$ decoupling for the twisted cubic (the moment curve $(t,t^2,t^3)$ in $\mathbb{R}^3$) and states that the results of the soft decoupling approach 'are suboptimal', suggesting his inequality as a possible tool for the $L^{10}$ moment problems. Glidewell (2026) verifies Demeter's conjecture at the $p=12$ endpoint for the moment curve in $\mathbb{R}^4$, noting that no estimates are known for $p<12$ in the truncated regime. On the zeta side, Bourgain's companion paper (Journal of the American Mathematical Society 30 (2017), 205-224) obtained the record pointwise bound $|\zeta(\frac{1}{2}+it)|\ll t^{13/84+\varepsilon}$ by a sixth-moment route that does not pass through $N_{10}$, and Bourgain's paper itself notes that its improvement (0.6) does not apply at the application's scale $\delta=1/N^2$; this remains so, and $13/84$ is still the best published pointwise exponent.

Problem Statement

The object is to sharpen the tenth-moment estimates for the two-scale exponential sums of Bourgain's 'Decoupling inequalities and some mean-value theorems', in the two respects the paper itself identifies. Both components serve one objective — sharp control of $N_{10}(\delta,\Delta)$ — and they are linked: the middle term of (4.18) is exactly what blocks the range extension in the second component.

Component 1 (reducing the middle term of (4.18)). Let $N_{10}(\delta,\Delta)$ denote the tenth moment over the unit box of the exponential sum $\sum_{k\sim N}e(kx_0+k^2x_1+\frac{1}{\delta}\phi_2(\frac{k}{N})x_2+\frac{1}{\Delta}\phi_3(\frac{k}{N})x_3)$, and assume the Wronskian nondegeneracy $W(\phi_2''',\phi_3''')\neq0$ and $\delta<\Delta$; this is Theorem 4's own generality, the paper's symbol $N_{10}(\delta,\Delta)$ (its (0.2)) being the special case $\phi_2(t)=t^{3/2}$, $\phi_3(t)=t^{1/2}$. Theorem 4 of the paper gives

$$N_{10}(\delta,\Delta)\ll\big[\delta\Delta^{3/4}N^7+(\delta+\Delta)N^6+N^5\big]N^{\varepsilon}.$$

In the regime $\Delta\gg\delta$ relevant to applications the term $(\delta+\Delta)N^6$ dominates. Prove an estimate of the same generality (same hypotheses, same definition of $N_{10}$) in which this middle term is reduced — for instance replaced by $(\delta+\Delta)N^{6-\theta}$ for some $\theta>0$, or by an expression subdominant to $\delta\Delta^{3/4}N^7+N^5$ when $\Delta\gg\delta$ — or prove that the middle term is close to sharp by exhibiting, for some admissible $\phi_2,\phi_3$ (the first-spacing phases $\phi_2(t)=t^{3/2}$, $\phi_3(t)=t^{1/2}$ included), parameters $\delta<\Delta$ and values of $N$ lying in the regime where the middle term dominates the other two terms of (4.18), that is $\delta\Delta^{3/4}N^7\ll(\delta+\Delta)N^6$ and $N^5\ll(\delta+\Delta)N^6$ up to $N^{\varepsilon}$ factors, with $N_{10}(\delta,\Delta)\gtrsim\Delta N^6$ up to $N^{\varepsilon}$ losses — $N_{10}$ being the unit-box moment defined above, so that the lower bound may be demonstrated by exhibiting a sub-box of the unit domain that carries the mass.

Component 2 (the true range of the $N^{5+\varepsilon}$ bound). For the first-spacing phases $\phi_2(t)=t^{3/2}$, $\phi_3(t)=t^{1/2}$ with $\Delta=\delta N$ (the paper's $N_{10}(\delta,\delta N)$, its (0.2)), determine for which $\delta$ the bound $N_{10}(\delta,\delta N)\ll N^{5+\varepsilon}$ holds. It is known to hold for $N^{-2}\le\delta\le N^{-33/18}$ (Theorem 9) and to fail for $\delta>N^{-11/7}$ (the Remark after Theorem 4, via the lower bound $\gtrsim\delta\Delta^{3/4}N^7$). The interval $N^{-33/18}<\delta\le N^{-11/7}$ is undetermined: extend the range in which the bound holds beyond $N^{-33/18}$, sharpen the failure threshold below $N^{-11/7}$, or identify the exact transition exponent between them.

The question concerns the quantities as defined in the paper rather than a redefined or moment-curve-only variant.

References

1. Jean Bourgain, Decoupling inequalities and some mean-value theorems, Journal d'Analyse Mathématique 133 (2017), 313-334. DOI: 10.1007/s11854-017-0035-2; arXiv:1406.7862. URL: https://doi.org/10.1007/s11854-017-0035-2
2. Jean Bourgain, Decoupling, exponential sums and the Riemann zeta function, Journal of the American Mathematical Society 30 (2017), 205-224. DOI: 10.1090/jams/860; arXiv:1408.5794. URL: https://doi.org/10.1090/jams/860
3. Jean Bourgain and Ciprian Demeter, Decouplings for curves and hypersurfaces with nonzero Gaussian curvature, Journal d'Analyse Mathématique 133 (2017), 279-311. DOI: 10.1007/s11854-017-0034-3. URL: https://doi.org/10.1007/s11854-017-0034-3
4. Ciprian Demeter, Larry Guth, and Hong Wang, Small cap decouplings, Geometric and Functional Analysis 30 (2020), 989-1062. DOI: 10.1007/s00039-020-00541-5; arXiv:1908.09166. URL: https://doi.org/10.1007/s00039-020-00541-5
5. Ciprian Demeter, On $L^{12}$ square root cancellation for exponential sums associated with nondegenerate curves in $\mathbb{R}^4$, Transactions of the American Mathematical Society 375 (2022), 7715-7746. DOI: 10.1090/tran/8677; arXiv:2101.08220. URL: https://doi.org/10.1090/tran/8677
6. Hongki Jung, A sharp $L^{10}$ decoupling for the twisted cubic, Journal d'Analyse Mathématique 149 (2023), 563-609. DOI: 10.1007/s11854-022-0258-8; arXiv:2011.10539. URL: https://doi.org/10.1007/s11854-022-0258-8
7. Jacob Glidewell, A Continuum of Small-cap Decouplings and Exponential Sums for the Moment Curve in $\mathbb{R}^4$, preprint, arXiv:2605.27065 (2026). URL: https://arxiv.org/abs/2605.27065
8. O. Robert and P. Sargos, Un théorème de moyenne pour les sommes d'exponentielles. Application à l'inégalité de Weyl, Publications de l'Institut Mathématique (Belgrade), Nouvelle Série 67(81) (2000), 14-30. Zbl: 1006.11046. URL: https://geodesic.mathdoc.fr/item/PIM_2000_N_S_67_81_a2/ (no DOI; also https://eudml.org/doc/122393)
9. E. Bombieri and H. Iwaniec, Some mean-value theorems for exponential sums, Annali della Scuola Normale Superiore di Pisa, Classe di Scienze (4) 13 (1986), 473-486. Zbl: 0615.10046. URL: https://eudml.org/doc/83988 (no DOI; also http://www.numdam.org/item?id=ASNSP_1986_4_13_3_473_0)
10. Larry Guth and James Maynard, New large value estimates for Dirichlet polynomials, Annals of Mathematics (2) 203 (2026), no. 2, 623-675. DOI: 10.4007/annals.2026.203.2.6; arXiv:2405.20552. URL: https://doi.org/10.4007/annals.2026.203.2.6