Let G be an abelian group, let S be a sequence of terms
s1β,s2β,...,snββG not all contained in a coset of a proper subgroup of
G, and let W be a sequence of n consecutive integers. Let WβS={w1βs1β+...+wnβsnβ:wiβatermofW,wiβξ =wjβforiξ =j},
which is a particular kind of weighted restricted sumset. We show that β£WβSβ£β₯min{β£Gβ£β1,n}, that WβS=G if nβ₯β£Gβ£+1, and also
characterize all sequences S of length β£Gβ£ with WβSξ =G. This
result then allows us to characterize when a linear equation
a1βx1β+...+arβxrββ‘Ξ±modn, where Ξ±,a1β,...,arββZ are
given, has a solution (x1β,...,xrβ)βZr modulo n with all xiβ
distinct modulo n. As a second simple corollary, we also show that there are
maximal length minimal zero-sum sequences over a rank 2 finite abelian group
Gβ Cn1βββCn2ββ (where n1ββ£n2β and n2ββ₯3) having k
distinct terms, for any kβ[3,min{n1β+1,exp(G)}]. Indeed, apart from
a few simple restrictions, any pattern of multiplicities is realizable for such
a maximal length minimal zero-sum sequence