Some Remarks on Vinogradov's Mean Value Theorem and Tarry's Problem.
On simultaneous additive equations II.
On exponential sums over smooth numbers.
Breaking classical convexity in Waring's problem: Sums of cubes and quasi-diagonal behaviour.
The application of a new mean value theorem to the fractional parts of polynomials
An explicit version of Birch's Theorem
Sums of three cubes, II
Estimates are provided for sth moments of cubic smooth Weyl sums, when 4 ≤ s ≤ 8, by enhancing the author's iterative method that delivers estimates beyond classical convexity. As a consequence, an improved lower bound is presented for the number of integers not exceeding X that are represented as the sum of three cubes of natural numbers.
The difficulty of the local solubility problem for additive equations
Artin's conjecture for septic and unidecic forms
Sums of fifth powers and related topics
Slim exceptional sets for sums of fourth and fifth powers
Pairs of cubic forms in many variables
Cubic moments of Fourier coefficients and pairs of diagonal quartic forms
We establish the non-singular Hasse principle for pairs of diagonal quartic equations in 22 or more variables. Our methods involve the estimation of a certain entangled two-dimensional 21st moment of quartic smooth Weyl sums via a novel cubic moment of Fourier coefficients.
Additive representation in thin sequences, I : Waring's problem for cubes
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