For k nin N, let f_k(boldsymbol alpha;X) be the Vinogradov exponential sum and, for a measurable set mathfrak B⊆[0,1)^k, define M_s,k(X;mathfrak B)=int_mathfrak B|f_k(boldsymbol alpha;X)|^2s, dboldsymbol alpha. Here mes(mathfrak B) denotes the measure of mathfrak B. The extended main conjecture. Suppose that k∈ N and mathfrak B⊆[0,1)^k is measurable. Whenever s∈ R satisfies s≥tfrac14k(k+1)+1, one has int_mathfrak B|f_k(boldsymbol alpha;X)|^2s, dboldsymbol alpha≪ X^varepsilon(X^smes(mathfrak B)+X^2s-k(k+1)/2). This extends the proved main conjecture for Vinogradov's mean value theorem from integration over the whole unit cube to arbitrary measurable subsets. The stated lower bound on s ensures suitable convergence of the singular series and singular integral; for smaller s, the expected secondary term may need modification.

The extended main conjecture in Vinogradov's mean value theorem

For $k\nin\mathbb N$ , let $f_k({\boldsymbol\alpha};X)$ be the Vinogradov exponential sum and, for a measurable set ${\mathfrak B}\subseteq[0,1)^k$ , define

M_{s,k}(X;{\mathfrak B})=\int_{\mathfrak B}|f_k({\boldsymbol\alpha};X)|^{2s}\,\mathrm d{\boldsymbol\alpha}.

Here $\mathrm{mes}({\mathfrak B})$ denotes the measure of ${\mathfrak B}$ .

The extended main conjecture. Suppose that $k\in\mathbb N$ and ${\mathfrak B}\subseteq[0,1)^k$ is measurable. Whenever $s\in\mathbb R$ satisfies $s\geq\tfrac14k(k+1)+1$ , one has

\int_{\mathfrak B}|f_k({\boldsymbol\alpha};X)|^{2s}\,\mathrm d{\boldsymbol\alpha}\ll X^\varepsilon\left(X^s\mathrm{mes}({\mathfrak B})+X^{2s-k(k+1)/2}\right).

This extends the proved main conjecture for Vinogradov's mean value theorem from integration over the whole unit cube to arbitrary measurable subsets. The stated lower bound on $s$ ensures suitable convergence of the singular series and singular integral; for smaller $s$ , the expected secondary term may need modification.

Equivalent formulations 1

Other statements of this same problem, merged from separate entries. Each is equivalent to the statement above — proving any one settles them all.

The extended main conjecture in Vinogradov's mean value theorem
Let $d\in\mathbb{N}$ , let $\boldsymbol{\alpha}=(\alpha_d,\ldots,\alpha_1)\in\mathbb{R}^d$ , and let $\mathfrak{D}\subseteq[0,1)^d$ be measurable. Write $d\boldsymbol{\alpha}=d\alpha_1\,d\alpha_2\cdots d\alpha_{d-1}\,d\alpha_d$ . For a measurable set $\mathfrak{D}$ , write $\operatorname{mes}(\mathfrak{D})$ for its measure. Suppose that $s$ is positive and that
$\operatorname{mes}(\mathfrak{D})\gg N^{1-d(d+1)/4}.$
The extended main conjecture. One has
$\int_{\mathfrak{D}}\left|\sum_{1\leq n\leq N}e\left(\sum_{1\leq i\leq d}\alpha_i n^i\right)\right|^{2s}d\boldsymbol{\alpha}\ll N^{\epsilon}\left(N^s\operatorname{mes}(\mathfrak{D})+N^{2s-d(d+1)/2}\right).$
The conjecture extends the main conjecture in Vinogradov's mean value theorem to measurable subsets of the coefficient space. It has been completely resolved by Bourgain, Demeter and Guth using decoupling inequalities for the moment curve, and by Wooley using efficient congruencing arguments.
source: Changkeun Oh and Kiseok Yeon, “An extended Vinogradov's mean value theorem”, arXiv:2506.01751 (2025).

Sources & referencesView supporting material

Primary source

Trevor D. Wooley, “Subconvexity in inhomogeneous Vinogradov systems”, arXiv:2202.14003 (2022).

analytic-number-theory exponential-sums vinogradov-mean-value-theorem

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The extended main conjecture in Vinogradov's mean value theorem

Equivalent formulations 1

The extended main conjecture in Vinogradov's mean value theorem

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