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The inverse problem for representation functions for general linear forms
The inverse problem for representation functions takes as input a triple
(X,f,L), where X is a countable semigroup, f : X --> N_0 \cup {\infty} a
function, L : a_1 x_1 + ... + a_h x_h an X-linear form and asks for a subset A
\subseteq X such that there are f(x) solutions (counted appropriately) to
L(x_1,...,x_h) = x for every x \in X, or a proof that no such subset exists.
This paper represents the first systematic study of this problem for
arbitrary linear forms when X = Z, the setting which in many respects is the
most natural one. Having first settled on the "right" way to count
representations, we prove that every primitive form has a unique representation
basis, i.e.: a set A which represents the function f \equiv 1. We also prove
that a partition regular form (i.e.: one for which no non-empty subset of the
coefficients sums to zero) represents any function f for which {f^{-1}(0)} has
zero asymptotic density. These two results answer questions recently posed by
Nathanson.
The inverse problem for partition irregular forms seems to be more
complicated. The simplest example of such a form is x_1 - x_2, and for this
form we provide some partial results. Several remaining open problems are
discussed.Comment: 15 pages, no figure
Isomorphisms in pro-categories
A morphism of a category which is simultaneously an epimorphism and a
monomorphism is called a bimorphism. In \cite{DR2} we gave characterizations of
monomorphisms (resp. epimorphisms) in arbitrary pro-categories, pro-(C), where
(C) has direct sums (resp. weak push-outs). In this paper we introduce the
notions of strong monomorphism and strong epimorphism. Part of their
significance is that they are preserved by functors. These notions and their
characterizations lead us to important classical properties and problems in
shape and pro-homotopy. For instance, strong epimorphisms allow us to give a
categorical point of view of uniform movability and to introduce a new kind of
movability, the sequential movability. Strong monomorphisms are connected to a
problem of K.Borsuk regarding a descending chain of retracts of ANRs. If (f: X
\to Y) is a bimorphism in the pointed shape category of topological spaces, we
prove that (f) is a weak isomorphism and (f) is an isomorphism provided (Y) is
sequentially movable and or is the suspension of a topological space.
If (f: X \to Y) is a bimorphism in the pro-category pro-(H_0) (consisting of
inverse systems in (H_0), the homotopy category of pointed connected CW
complexes) we show that (f) is an isomorphism provided (Y) is sequentially
movable.Comment: to appear in the Journal of Pure and Applied Algebr
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