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A nonstandard technique in combinatorial number theory
In [9], [15] it has been introduced a technique, based on nonstandard
analysis, to study some problems in combinatorial number theory. In this paper
we present three applications of this technique: the first one is a new proof
of a known result regarding the algebra of \betaN, namely that the center of
the semigroup (\beta\mathbb{N};\oplus) is \mathbb{N}; the second one is a
generalization of a theorem of Bergelson and Hindman on arithmetic progressions
of lenght three; the third one regards the partition regular polynomials in
Z[X], namely the polynomials in Z[X] that have a monochromatic solution for
every finite coloration of N. We will study this last application in more
detail: we will prove some algebraical properties of the sets of such
polynomials and we will present a few examples of nonlinear partition regular
polynomials. In the first part of the paper we will recall the main results of
the nonstandard technique that we want to use, which is based on a
characterization of ultrafilters by means of nonstandard analysis
Partition Regularity of Nonlinear Polynomials: a Nonstandard Approach
In 2011, Neil Hindman proved that for every natural number the
polynomial \begin{equation*} \sum_{i=1}^{n} x_{i}-\prod\limits_{j=1}^{m} y_{j}
\end{equation*} has monochromatic solutions for every finite coloration of
. We want to generalize this result to two classes of nonlinear
polynomials. The first class consists of polynomials
of the following kind: \begin{equation*}
P(x_{1},...,x_{n},y_{1},...,y_{m})=\sum_{i=1}^{n}a_{i}x_{i}M_{i}(y_{1},...,y_{m}),
\end{equation*} where are natural numbers,
has monochromatic solutions for every finite
coloration of and the degree of each variable in
is at most one. An example of such a polynomial is
\begin{equation*} x_{1}y_{1}+x_{2}y_{1}y_{2}-x_{3}.\end{equation*} The second
class of polynomials generalizing Hindman's result is more complicated to
describe; its particularity is that the degree of some of the involved
variables can be greater than one.\\ The technique that we use relies on an
approach to ultrafilters based on Nonstandard Analysis. Perhaps, the most
interesting aspect of this technique is that, by carefully chosing the
appropriate nonstandard setting, the proof of the main results can be obtained
by very simple algebraic considerations
Hypernatural Numbers as Ultrafilters
In this paper we present a use of nonstandard methods in the theory of
ultrafilters and in related applications to combinatorics of numbers
Nonstandard Methods in Ramsey Theory and Combinatorial Number Theory
The goal of this present manuscript is to introduce the reader to the
nonstandard method and to provide an overview of its most prominent
applications in Ramsey theory and combinatorial number theory.Comment: 126 pages. Comments welcom
Fermat, Leibniz, Euler, and the gang: The true history of the concepts of limit and shadow
Fermat, Leibniz, Euler, and Cauchy all used one or another form of
approximate equality, or the idea of discarding "negligible" terms, so as to
obtain a correct analytic answer. Their inferential moves find suitable proxies
in the context of modern theories of infinitesimals, and specifically the
concept of shadow. We give an application to decreasing rearrangements of real
functions.Comment: 35 pages, 2 figures, to appear in Notices of the American
Mathematical Society 61 (2014), no.
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