Arithmetic-progression-weighted subsequence sums

作者：Grynkiewicz David J^*; Philipp Andreas; Ponomarenko Vadim

来源：Israel Journal of Mathematics, 2013, 193(1): 359-398.

DOI：10.1007/s11856-012-0119-8

Let G be an abelian group, let s be a sequence of terms s(1), s(2),center dot center dot center dot, s(n) epsilon G not all contained in a coset of a proper subgroup of G, and let W be a sequence of n consecutive integers. Let
W circle dot S={w(1)s(1)+center dot center dot center dot+w(n)s(n):w(i) a term of W, w(i) not equal w(j) for i not equal j},
which is a particular kind of weighted restricted sumset. We show that vertical bar W circle dot S vertical bar >= min{vertical bar G vertical bar - 1, n}, that W circle dot S = G if n >= vertical bar G vertical bar + 1, and also characterize all sequences S of length vertical bar G vertical bar with W circle dot S not equal G. This result then allows us to characterize when a linear equation
a(1)x(1)+center dot center dot center dot+a(r)x(r) alpha mod n,
where alpha, a(1),center dot center dot center dot, a(r) epsilon Z are given, has a solution (x(1),center dot center dot center dot, x(r)) epsilon Z(r) modulo n with all x(i) 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 G congruent to C-n1 circle plus C-n2 (where n(1) vertical bar n(2) and n(2) >= 3) having k distinct terms, for any k epsilon [3, min{n(1) + 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.

• 出版日期2013-1