9 research outputs found
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
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
A note on maximal progression-free sets
AbstractErdős et al [Greedy algorithm, arithmetic progressions, subset sums and divisibility, Discrete Math. 200 (1999) 119–135.] asked whether there exists a maximal set of positive integers containing no three-term arithmetic progression and such that the difference of its adjacent elements approaches infinity. This note answers the question affirmatively by presenting such a set in which the difference of adjacent elements is strictly increasing. The construction generalizes to arithmetic progressions of any finite length