68 research outputs found
Detailed Structure for Freiman's 3k-3 Theorem
Let A be a finite set of integers. We prove that if |A| is at least 2 and
|A+A| is 3|A|-3, then one of the following is true:
1. A is a bi-arithmetic progression;
2. A+A contains an arithmetic progression of length 2|A|-1;
3. |A| is 6 and A is Freiman isomorphic to the set
{(0,0),(0,1),(0,2),(1,0),(1,1),(2,0)};
4. A is Freiman isomorphic to a set in either the form of {0,2,...,2k} union
B union {n} for some non-negative integer k at most n/2 -2 or the form of {0}
union C union D union {n}, where n=2|A|-2, B is left dense in [2k,n-1], C is
right dense in [1,u] for some u in [4,n-6], D is left dense in [u+2,n-1], B,C,D
are anti-symmetric and additively minimal in the correspondent host intervals
Planting Kurepa trees and killing Jech-Kunen trees in a model by using one inaccessible cardinal
By an omega_1--tree we mean a tree of power omega_1 and height omega_1. Under
CH and 2^{omega_1}> omega_2 we call an omega_1--tree a Jech--Kunen tree if it
has kappa many branches for some kappa strictly between omega_1 and
2^{omega_1}. In this paper we prove that, assuming the existence of one
inaccessible cardinal,
(1) it is consistent with CH plus 2^{omega_1}> omega_2 that there exist
Kurepa trees and there are no Jech--Kunen trees,
(2) it is consistent with CH plus 2^{omega_1}= omega_4 that only Kurepa trees
with omega_3 many branches exist
Abstract densities and ideals of sets
Abstract upper densities are monotone and subadditive functions from the
power set of positive integers to the unit real interval that generalize the
upper densities used in number theory, including the upper asymptotic density,
the upper Banach density, and the upper logarithmic density. We answer a
question posed by G. Grekos in 2013, and prove the existence of translation
invariant abstract upper densities onto the unit interval, whose null sets are
precisely the family of finite sets, or the family of sequences whose series of
reciprocals converge. We also show that no such density can be atomless. (More
generally, these results also hold for a large class of summable ideals.
Characterizing the structure of A+B when A+B has small upper Banach density
AbstractLet A and B be two infinite sets of non-negative integers. Similar to Kneser's Theorem (Theorem 1.1 below) we characterize the structure of A+B when the upper Banach density of A+B is less than the sum of the upper Banach density of A and the upper Banach density of B
An integer construction of infinitesimals: Toward a theory of Eudoxus hyperreals
A construction of the real number system based on almost homomorphisms of the
integers Z was proposed by Schanuel, Arthan, and others. We combine such a
construction with the ultrapower or limit ultrapower construction, to construct
the hyperreals out of integers. In fact, any hyperreal field, whose universe is
a set, can be obtained by such a one-step construction directly out of
integers. Even the maximal (i.e., On-saturated) hyperreal number system
described by Kanovei and Reeken (2004) and independently by Ehrlich (2012) can
be obtained in this fashion, albeit not in NBG. In NBG, it can be obtained via
a one-step construction by means of a definable ultrapower (modulo a suitable
definable class ultrafilter).Comment: 17 pages, 1 figur
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