Slim exceptional sets and the asymptotic formula in Waring's problem

By avoiding a conventional application of Bessel’s inequality in favour of explicitly controlling an exponential sum over the exceptional set, in our previous work [9, 10], we have exploited additional variables so as to enhance exceptional set estimates in various additive problems of Waring type. In this paper we turn to the problem of establishing the expected asymptotic formula in Waring’s problem. The methods introduced herein, although discussed in the context of the asymptotic formula, should nonetheless provide a useful model for future excursions involving exceptional sets in additive problems.