33,414 research outputs found
Sets Characterized by Missing Sums and Differences in Dilating Polytopes
A sum-dominant set is a finite set of integers such that .
As a typical pair of elements contributes one sum and two differences, we
expect sum-dominant sets to be rare in some sense. In 2006, however, Martin and
O'Bryant showed that the proportion of sum-dominant subsets of
is bounded below by a positive constant as . Hegarty then extended
their work and showed that for any prescribed , the
proportion of subsets of that are missing
exactly sums in and exactly differences in
also remains positive in the limit.
We consider the following question: are such sets, characterized by their
sums and differences, similarly ubiquitous in higher dimensional spaces? We
generalize the integers in a growing interval to the lattice points in a
dilating polytope. Specifically, let be a polytope in with
vertices in , and let now denote the proportion of
subsets of that are missing exactly sums in and
exactly differences in . As it turns out, the geometry of
has a significant effect on the limiting behavior of . We define
a geometric characteristic of polytopes called local point symmetry, and show
that is bounded below by a positive constant as if
and only if is locally point symmetric. We further show that the proportion
of subsets in that are missing exactly sums and at least
differences remains positive in the limit, independent of the geometry of .
A direct corollary of these results is that if is additionally point
symmetric, the proportion of sum-dominant subsets of also remains
positive in the limit.Comment: Version 1.1, 23 pages, 7 pages, fixed some typo
Partition Problems and a Pattern of Vertical Sums
We give a possible explanation for the mystery of a missing number in the
statement of a problem that asks for the non-negative integers to be
partitioned into three subsets. We interpret the missing number as one of the
clues that can lead to a more standard solution to the problem, using only
congruence modulo five, and we give the details to the new solution, which is
based on an algorithm inspired by noticing alternating differences between sums
of elements of the same rank in the three sets. Our new solution is equivalent
to the partition consisting of numbers with remainders one or three modulo
five, two or four modulo five, and multiples of five, which we call the
standard partition. We then find all other similar statements with the same
pattern of sums, we apply the algorithm to them, and we describe all the
partitions obtained, up to a certain equivalence. There are different
such statements, they produce twenty different partitions (other than the
standard one) whose sets of the first five columns are not permutations of each
other, and only one of them (the one produced by the original statement of the
problem we study) is equivalent to the standard partition. Finally, we
construct infinitely many partitions equivalent to the standard one, and we
give a possible generalization and a sample partition problem asking for the
non-negative integers to be partitioned into four sets.Comment: 18 pages, 6 figure
Sets Characterized by Missing Sums and Differences
A more sums than differences (MSTD) set is a finite subset S of the integers
such |S+S| > |S-S|. We show that the probability that a uniform random subset
of {0, 1, ..., n} is an MSTD set approaches some limit rho > 4.28 x 10^{-4}.
This improves the previous result of Martin and O'Bryant that there is a lower
limit of at least 2 x 10^{-7}. Monte Carlo experiments suggest that rho \approx
4.5 \x 10^{-4}. We present a deterministic algorithm that can compute rho up to
arbitrary precision. We also describe the structure of a random MSTD subset S
of {0, 1, ..., n}. We formalize the intuition that fringe elements are most
significant, while middle elements are nearly unrestricted. For instance, the
probability that any ``middle'' element is in S approaches 1/2 as n ->
infinity, confirming a conjecture of Miller, Orosz, and Scheinerman. In
general, our results work for any specification on the number of missing sums
and the number of missing differences of S, with MSTD sets being a special
case.Comment: 32 pages, 1 figure, 1 tabl
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
Explicit constructions of infinite families of MSTD sets
We explicitly construct infinite families of MSTD (more sums than
differences) sets. There are enough of these sets to prove that there exists a
constant C such that at least C / r^4 of the 2^r subsets of {1,...,r} are MSTD
sets; thus our family is significantly denser than previous constructions
(whose densities are at most f(r)/2^{r/2} for some polynomial f(r)). We
conclude by generalizing our method to compare linear forms epsilon_1 A + ... +
epsilon_n A with epsilon_i in {-1,1}.Comment: Version 2: 14 pages, 1 figure. Includes extensions to ternary forms
and a conjecture for general combinations of the form Sum_i epsilon_i A with
epsilon_i in {-1,1} (would be a theorem if we could find a set to start the
induction in general
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