Vinogradov’s mean value theorem via efficient congruencing


We obtain estimates for Vinogradov’s integral that for the first time approach those conjectured to be the best possible. Several applications of these new bounds are provided. In particular, the conjectured asymptotic formula in Waring’s problem holds for sums of $s$ $k$th powers of natural numbers whenever $s\ge 2k^2+2k-3$.


Trevor D. Wooley

School of Mathematics, University of Bristol, University Walk, Bristol BS8 1TW, United Kingdom