# Nested efficient congruencing and relatives of Vinogradov's mean value theorem

@article{Wooley2018NestedEC,
title={Nested efficient congruencing and relatives of Vinogradov's mean value theorem},
author={Trevor D. Wooley},
journal={Proceedings of the London Mathematical Society},
year={2018},
volume={118}
}
• T. Wooley
• Published 3 August 2017
• Mathematics
• Proceedings of the London Mathematical Society
We apply a nested variant of multigrade efficient congruencing to estimate mean values related to that of Vinogradov. We show that when φj∈Z[t] (1⩽j⩽k) is a system of polynomials with non‐vanishing Wronskian, and s⩽k(k+1)/2 , then for all complex sequences (an) , and for each ε>0 , one has ∫[0,1)k∑|n|⩽Xane(α1φ1(n)+⋯+αkφk(n))2sdα≪Xε∑|n|⩽X|an|2s.As a special case of this result, we confirm the main conjecture in Vinogradov's mean value theorem for all exponents k , recovering the recent…
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