8,260 research outputs found
Classification theorems for sumsets modulo a prime
Let be the finite field of prime order and be a subsequence
of . We prove several classification results about the following
questions: (1) When can one represent zero as a sum of some elements of ?
(2) When can one represent every element of as a sum of some elements
of ? (3) When can one represent every element of as a sum of
elements of ?Comment: 35 pages, to appear in JCT
Long -zero-free sequences in finite cyclic groups
A sequence in the additive group of integers modulo is
called -zero-free if it does not contain subsequences with length and
sum zero. The article characterizes the -zero-free sequences in of length greater than . The structure of these sequences is
completely determined, which generalizes a number of previously known facts.
The characterization cannot be extended in the same form to shorter sequence
lengths. Consequences of the main result are best possible lower bounds for the
maximum multiplicity of a term in an -zero-free sequence of any given length
greater than in , and also for the combined
multiplicity of the two most repeated terms. Yet another application is finding
the values in a certain range of a function related to the classic theorem of
Erd\H{o}s, Ginzburg and Ziv.Comment: 11 page
Long zero-free sequences in finite cyclic groups
A sequence in an additively written abelian group is called zero-free if each
of its nonempty subsequences has sum different from the zero element of the
group. The article determines the structure of the zero-free sequences with
lengths greater than in the additive group \Zn/ of integers modulo .
The main result states that for each zero-free sequence of
length in \Zn/ there is an integer coprime to such that if
denotes the least positive integer in the congruence class
(modulo ), then . The answers to a number of
frequently asked zero-sum questions for cyclic groups follow as immediate
consequences. Among other applications, best possible lower bounds are
established for the maximum multiplicity of a term in a zero-free sequence with
length greater than , as well as for the maximum multiplicity of a
generator. The approach is combinatorial and does not appeal to previously
known nontrivial facts.Comment: 13 page
Arithmetic-Progression-Weighted Subsequence Sums
Let be an abelian group, let be a sequence of terms
not all contained in a coset of a proper subgroup of
, and let be a sequence of consecutive integers. Let
which is a particular kind of weighted restricted sumset. We show that , that if , and also
characterize all sequences of length with . This
result then allows us to characterize when a linear equation
where are
given, has a solution modulo with all
distinct modulo . As a second simple corollary, we also show that there are
maximal length minimal zero-sum sequences over a rank 2 finite abelian group
(where and ) having
distinct terms, for any . Indeed, apart from
a few simple restrictions, any pattern of multiplicities is realizable for such
a maximal length minimal zero-sum sequence
On n-sum of an abelian group of order n
Let be an additive finite abelian group of order , and let be a
sequence of elements in , where . Suppose that contains
distinct elements. Let denote the set that consists of all
elements in which can be expressed as the sum over a subsequence of length
. In this paper we prove that, either or This confirms a conjecture by Y.O. Hamidoune in 2000
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