7,353 research outputs found
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
Counting Co-Cyclic Lattices
There is a well-known asymptotic formula, due to W. M. Schmidt (1968) for the
number of full-rank integer lattices of index at most in .
This set of lattices can naturally be partitioned with respect to the
factor group . Accordingly, we count the number of full-rank
integer lattices such that is
cyclic and of order at most , and deduce that these co-cyclic lattices are
dominant among all integer lattices: their natural density is . The problem is motivated by
complexity theory, namely worst-case to average-case reductions for lattice
problems
Counting MSTD Sets in Finite Abelian Groups
In an abelian group G, a more sums than differences (MSTD) set is a subset A
of G such that |A+A|>|A-A|. We provide asymptotics for the number of MSTD sets
in finite abelian groups, extending previous results of Nathanson. The proof
contains an application of a recently resolved conjecture of Alon and Kahn on
the number of independent sets in a regular graph.Comment: 17 page
Number fields with prescribed norms
We study the distribution of extensions of a number field with fixed
abelian Galois group , from which a given finite set of elements of are
norms. In particular, we show the existence of such extensions. Along the way,
we show that the Hasse norm principle holds for of -extensions of
, when ordered by conductor. The appendix contains an alternative purely
geometric proof of our existence result.Comment: 35 pages, comments welcome
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