The cubic case of the main conjecture in Vinogradov's mean value theorem

Abstract

We apply a variant of the multigrade efficient congruencing method to estimate Vinogradov's integral of degree $3$ for moments of order $2s$, establishing strongly diagonal behaviour for $1\le s\le 6$. Consequently, the main conjecture is now known to hold for the first time in a case of degree exceeding $2$.

arXiv commentarXiv admin note: substantial text overlap with arXiv:1401.2932

References in corpus (6)

• Vinogradov's mean value theorem via efficient congruencing
• Translation invariance, exponential sums, and Waring's problem
• Multigrade efficient congruencing and Vinogradov's mean value theorem
• Mean value estimates for odd cubic Weyl sums
• Rational solutions of pairs of diagonal equations, one cubic and one quadratic
• Subconvexity for additive equations: pairs of undenary cubic forms

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• Discrete Fourier restriction via efficient congruencing: basic principles
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• Improved ${\ell }^p$-Boundedness for Integral $k$-Spherical Maximal Functions
• Reversing a philosophy: from counting to square functions and decoupling
• A short proof of ${\ell }^2$ decoupling for the moment curve
• Efficient congruencing in ellipsephic sets: the quadratic case
• Approximating the main conjecture in Vinogradov's mean value theorem
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• The Cubic Case of Vinogradov's Mean Value Theorem --- A Simplified Approach to Wooley's "Efficient Congruencing"
• A New $k$-th Derivative Estimate for Exponential Sums via Vinogradov's Mean Value
• Efficient congruencing in ellipsephic sets: the general case
• Arithmetic of higher-dimensional orbifolds and a mixed Waring problem
• Discrete restriction for $(x,x^3)$ and related topics
• Hybrid bounds on two-parametric family Weyl sums along smooth curves
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• Simultaneous additive equations: Repeated and differing degrees
• Restricted mean value theorems and metric theory of restricted Weyl sums
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• Small cap decoupling for the moment curve in ${\mathbb{R}}^3$
• On a Hybrid Version of the Vinogradov Mean Value Theorem
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• On digits of Mersenne numbers
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• Hausdorff dimension of the large values of Weyl sums
• Large sieve inequalities with power moduli and Waring's problem
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• Fourier decay of fractal measures on surfaces of co-dimension two in ${\mathbb{R}}^5$
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• Supercritical discrete restriction estimates for forms in many variables
• Weyl sums with multiplicative coefficients and joint equidistribution