Subset sums over Galois rings

Let p be an odd prime and R = GR(p(2), p(2r)) be a Galois Ring of characteristic p(2) with cardinality p(2r). For a subset D of R, let N-D(n, b) = #{S subset of D vertical bar Sigma(x is an element of s) x = b and vertical bar S vertical bar = n} and R(k) = {y is an element of R vertical bar y = x(k) for some x is an element of R} We give the asymptotic formula of N-D (n, b) for D = R(k) \ {0}. From which we deduce that any element in R is the sum of n different k-th power under sufficient condition. (C) 2019 Elsevier Inc. All rights reserved.