Near-optimal mean value estimates for multidimensional Weyl sums
  We obtain sharp estimates for multidimensional generalisations of Vinogradov's mean value theorem for arbitrary translation-dilation invariant systems, achieving constraints on the number of variables approaching those conjectured to be the best possible. Several applications of our bounds are discussed. 
Scott T. Parsell, Sean M. Prendiville and Trevor D. Wooley
math.NT