EXCEPTIONAL SETS IN WARING’S PROBLEM: TWO SQUARES, TWO CUBES AND TWO SIXTH POWERS

LetR(n) denote the number of representations of a large positive integernas the sum of two squares, two cubes and two sixth powers. In this paper, it is proved that the anticipated asymptotic formula ofR(n) fails for at mostO((logX)2+ε ) positive integers not exceedingX. This is an improvement of T. D. Wooley’s result which requiresO((logX)3+ε ). 2010Mathematics Subject Classification: 11P05, 11P55, 11N37. Key words and phrases: Waring’s problem, Hardy-Littlewood method, Exceptional sets, Asymptotic formula.