On the asymptotic formula in Waring's problem with shifts

We show that for integers k≥4 and s≥k2+(3k−1)/4, we have an asymptotic formula for the number of solutions to the inequality |(x1−θ1)k+…+(xs−θs)k−τ|<η in positive integers xi, where θi∈(0,1) with θ1 irrational, η∈(0,1], and τ>0 is sufficiently large. We use Freeman's variant of the Davenport-Heilbronn method, along with a new estimate on the Hardy-Littlewood minor arcs, to obtain this improvement on the original result of Chow.