84 research outputs found
Infinite co-minimal pairs in the integers and integral lattices
Given two nonempty subsets of a group , they are said to form a
co-minimal pair if , and for any
and for any
. In this article, we show several new results
on co-minimal pairs in the integers and the integral lattices. We prove that
for any , the group admits infinitely many
automorphisms such that for each such automorphism , there exists a
subset of such that and form a co-minimal
pair. The existence and construction of co-minimal pairs in the integers with
both the subsets and () of infinite cardinality was unknown.
We show that such pairs exist and explicitly construct these pairs satisfying a
number of algebraic properties
On minimal complements in groups
Let be nonempty subsets in an arbitrary group . The set
is said to be a complement to if and it is minimal if no
proper subset of is a complement to . We show that, if is finite
then every complement of has a minimal complement, answering a problem of
Nathanson. This also shows the existence of minimal -nets for every
in finitely generated groups. Further, we give necessary and
sufficient conditions for the existence of minimal complements of a certain
class of infinite subsets in finitely generated abelian groups, partially
answering another problem of Nathanson. Finally, we provide infinitely many
examples of infinite subsets of abelian groups of arbitrary finite rank
admitting minimal complements.Comment: Minor correction
The density of ramified primes
Let be a number field, be a domain with fraction field
of characteristic zero and be a representation such that
is semisimple. If admits a
finite monomorphism from a power series ring with coefficients in a -adic
integer ring (resp. is an affinoid algebra over a -adic number
field) and is continuous with respect to the maximal ideal adic topology
(resp. the Banach algebra topology), then we prove that the set of ramified
primes of is of density zero. If is a complete local
Noetherian ring over with finite residue field of characteristic
, is continuous with respect to the maximal ideal adic topology and
the kernels of pure specializations of form a Zariski-dense subset of
, then we show that the set of ramified primes of
is of density zero. These results are analogues, in the context of big
Galois representations, of a result of Khare and Rajan, and are proved relying
on their result
Variation of Weyl modules in -adic families
Given a Weil-Deligne representation with coefficients in a domain, we prove
the rigidity of the structures of the Frobenius-semisimplifications of the Weyl
modules associated to its pure specializations. Moreover, we show that the
structures of the Frobenius-semisimplifications of the Weyl modules attached to
a collection of pure representations are rigid if these pure representations
lift to Weil-Deligne representations over domains containing a domain
and a pseudorepresentation over parametrizes the
traces of these lifts.Comment: 6 pages, comments welcom
Conductors in -adic families
Given a Weil-Deligne representation of the Weil group of an -adic
number field with coefficients in a domain , we show that its pure
specializations have the same conductor. More generally, we prove that the
conductors of a collection of pure representations are equal if they lift to
Weil-Deligne representations over domains containing and the
traces of these lifts are parametrized by a pseudorepresentation over
.Comment: arXiv admin note: text overlap with arXiv:1410.384
Minimal additive complements in finitely generated abelian groups
Given two non-empty subsets in an arbitrary abelian group
, is said to be an additive complement to if and it is
minimal if no proper subset of is a complement to . The notion was
introduced by Nathanson and previous work by him, Chen--Yang,
Kiss--S\`andor--Yang etc. focussed on . In the higher rank case,
recent work by the authors treated a class of subsets, namely the eventually
periodic sets. However, for infinite subsets, not of the above type, the
question of existence or inexistence of minimal complements is open. In this
article, we study subsets which are not eventually periodic. We introduce the
notion of "spiked subsets" and give necessary and sufficient conditions for the
existence of minimal complements for them. This provides a partial answer to a
problem of Nathanson.Comment: 25 pages, 8 figure
Infinite co-minimal pairs involving lacunary sequences and generalisations to higher dimensions
The study of minimal complements in a group or a semigroup was initiated by
Nathanson. The notion of minimal complements and being a minimal complement
leads to the notion of co-minimal pairs which was considered in a prior work of
the authors. In this article, we study which type of subsets in the integers
and free abelian groups of higher rank can be a part of a co-minimal pair. We
show that a majority of lacunary sequences have this property. From the
conditions established, one can show that any infinite subset of any finitely
generated abelian group has uncountably many subsets which is a part of a
co-minimal pair. Further, the uncountable collection of sets can be chosen so
that they satisfy certain algebraic properties
A Cheeger type inequality in finite Cayley sum graphs
Let be a finite group and be a symmetric generating set of with
. We show that if the undirected Cayley sum graph is
an expander graph and is non-bipartite, then the spectrum of its normalised
adjacency operator is bounded away from . We also establish an explicit
lower bound for the spectrum of these graphs, namely, the non-trivial
eigenvalues of the normalised adjacency operator lies in the interval
, where
denotes the (vertex) Cheeger constant of the -regular graph
and . Further, we improve upon a recently
obtained bound on the non-trivial spectrum of the normalised adjacency operator
of the non-bipartite Cayley graph .Comment: Grant number adde
Asymptotic complements in the integers
Let be a non-empty subset of the integers. A nonempty
set is said to be an asymptotic complement to if
contains almost all the integers except a set of finite size. is said
to be a minimal asymptotic complement if is an asymptotic complement, but
is not an asymptotic complement .
Asymptotic complements have been studied in the context of representations of
integers since the time of Erd\H{o}s, Hanani, Lorentz and others, while the
notion of minimal asymptotic complements is due to Nathanson. In this article,
we study minimal asymptotic complements in and deal with a problem
of Nathanson on their existence and their inexistence.Comment: Final version, to appear in the Journal of Number Theor
On non-minimal complements
The notion of minimal complements was introduced by Nathanson in 2011. Since
then, the existence or the inexistence of minimal complements of sets have been
extensively studied. Recently, the study of inverse problems, i.e., which sets
can or cannot occur as minimal complements has gained traction. For example,
the works of Kwon, Alon--Kravitz--Larson, Burcroff--Luntzlara and also that of
the authors, shed light on some of the questions in this direction. These works
have focussed mainly on the group of integers, or on abelian groups. In this
work, our motivation is two-fold: (i) to show some new results on the inverse
problem, (ii) to concentrate on the inverse problem in not necessarily abelian
groups. As a by-product, we obtain new results on non-minimal complements in
the group of integers and more generally, in any finitely generated abelian
group of positive rank and in any free abelian group of positive rank.
Moreover, we show the existence of uncountably many subsets in such groups
which are "robust" non-minimal complements
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