865 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
A Study on Integer Additive Set-Graceful Graphs
A set-labeling of a graph is an injective function , where is a finite set and a set-indexer of is a
set-labeling such that the induced function defined by
for every is also injective. An integer additive set-labeling is
an injective function ,
is the set of all non-negative integers and an integer additive
set-indexer is an integer additive set-labeling such that the induced function
defined by is also injective. In this paper, we extend the concepts of set-graceful
labeling to integer additive set-labelings of graphs and provide some results
on them.Comment: 11 pages, submitted to JARP
Most Subsets are Balanced in Finite Groups
The sumset is one of the most basic and central objects in additive number
theory. Many of the most important problems (such as Goldbach's conjecture and
Fermat's Last theorem) can be formulated in terms of the sumset of a set of integers . A finite set of integers is
sum-dominated if . Though it was believed that the percentage of
subsets of that are sum-dominated tends to zero, in 2006 Martin
and O'Bryant proved a very small positive percentage are sum-dominated if the
sets are chosen uniformly at random (through work of Zhao we know this
percentage is approximately ). While most sets are
difference-dominated in the integer case, this is not the case when we take
subsets of many finite groups. We show that if we take subsets of larger and
larger finite groups uniformly at random, then not only does the probability of
a set being sum-dominated tend to zero but the probability that
tends to one, and hence a typical set is balanced in this case. The cause of
this marked difference in behavior is that subsets of have a
fringe, whereas finite groups do not. We end with a detailed analysis of
dihedral groups, where the results are in striking contrast to what occurs for
subsets of integers.Comment: Version 2.0, 11 pages, 2 figure
When almost all sets are difference dominated
We investigate the relationship between the sizes of the sum and difference
sets attached to a subset of {0,1,...,N}, chosen randomly according to a
binomial model with parameter p(N), with N^{-1} = o(p(N)). We show that the
random subset is almost surely difference dominated, as N --> oo, for any
choice of p(N) tending to zero, thus confirming a conjecture of Martin and
O'Bryant. The proofs use recent strong concentration results.
Furthermore, we exhibit a threshold phenomenon regarding the ratio of the
size of the difference- to the sumset. If p(N) = o(N^{-1/2}) then almost all
sums and differences in the random subset are almost surely distinct, and in
particular the difference set is almost surely about twice as large as the
sumset. If N^{-1/2} = o(p(N)) then both the sum and difference sets almost
surely have size (2N+1) - O(p(N)^{-2}), and so the ratio in question is almost
surely very close to one. If p(N) = c N^{-1/2} then as c increases from zero to
infinity (i.e., as the threshold is crossed), the same ratio almost surely
decreases continuously from two to one according to an explicitly given
function of c.
We also extend our results to the comparison of the generalized difference
sets attached to an arbitrary pair of binary linear forms. For certain pairs of
forms f and g, we show that there in fact exists a sharp threshold at c_{f,g}
N^{-1/2}, for some computable constant c_{f,g}, such that one form almost
surely dominates below the threshold, and the other almost surely above it.
The heart of our approach involves using different tools to obtain strong
concentration of the sizes of the sum and difference sets about their mean
values, for various ranges of the parameter p.Comment: Version 2.1. 24 pages. Fixed a few typos, updated reference
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