19 research outputs found
Combinatorial and Additive Number Theory Problem Sessions: '09--'19
These notes are a summary of the problem session discussions at various CANT
(Combinatorial and Additive Number Theory Conferences). Currently they include
all years from 2009 through 2019 (inclusive); the goal is to supplement this
file each year. These additions will include the problem session notes from
that year, and occasionally discussions on progress on previous problems. If
you are interested in pursuing any of these problems and want additional
information as to progress, please email the author. See
http://www.theoryofnumbers.com/ for the conference homepage.Comment: Version 3.4, 58 pages, 2 figures added 2019 problems on 5/31/2019,
fixed a few issues from some presenters 6/29/201
The N-eigenvalue Problem and Two Applications
We consider the classification problem for compact Lie groups
which are generated by a single conjugacy class with a fixed number of
distinct eigenvalues. We give an explicit classification when N=3, and apply
this to extract information about Galois representations and braid group
representations.Comment: 30 pages. version 3: many typos fixed, section 6 substantially
reorganized. To appear in Int. Math. Res. No
Probabilistic and extremal studies in additive combinatorics
The results in this thesis concern extremal and probabilistic topics in number theoretic settings.
We prove sufficient conditions on when certain types of integer solutions to linear systems of
equations in binomial random sets are distributed normally, results on the typical approximate
structure of pairs of integer subsets with a given sumset cardinality, as well as upper bounds
on how large a family of integer sets defining pairwise distinct sumsets can be. In order to
prove the typical structural result on pairs of integer sets, we also establish a new multipartite
version of the method of hypergraph containers, generalizing earlier work by Morris, Saxton
and Samotij.L'objectiu de la combinatòria additiva “històricament també anomenada teoria combinatòria de nombres” és la d’estudiar l'estructura additiva de conjunts en determinats grups ambient. La combinatòria extremal estudia quant de gran pot ser una col·lecció d'objectes finits abans d'exhibir determinats requisits estructurals. La combinatòria probabilística analitza estructures combinatòries aleatòries, identificant en particular l'estructura dels objectes combinatoris típics. Entre els estudis més celebrats hi ha el treball de grafs aleatoris iniciat per Erdös i Rényi. Un exemple especialment rellevant de com aquestes tres àrees s'entrellacen és el desenvolupament per Erdös del mètode probabilístic en teoria de nombres i en combinatòria, que mostra l'existència de moltes estructures extremes en configuracions additives utilitzant tècniques probabilistes. Tots els temes d'aquesta tesi es troben en la intersecció d'aquestes tres àrees, i apareixen en els problemes següents. Solucions enteres de sistemes d'equacions lineals. Els darrers anys s'han obtingut resultats pel que fa a l’existència de llindars per a determinades solucions enteres a un sistema arbitrari d'equacions lineals donat, responent a la pregunta de quan s'espera que el subconjunt aleatori binomial d'un conjunt inicial de nombres enters contingui solucions gairebé sempre. La següent pregunta lògica és la següent. Suposem que estem en la zona en que hi haurà solucions enteres en el conjunt aleatori binomial, com es distribueixen aleshores aquestes solucions? Al capítol 1, avançarem per respondre aquesta pregunta proporcionant condicions suficients per a quan una gran varietat de solucions segueixen una distribució normal. També parlarem de com, en determinats casos, aquestes condicions suficients també són necessàries. Conjunts amb suma acotada. Què es pot dir de l'estructura de dos conjunts finits en un grup abelià si la seva suma de Minkowski no és molt més gran que la dels conjunts? Un resultat clàssic de Kneser diu que això pot passar si i només si la suma de Minkowski és periòdica respecte a un subgrup propi. En el capítol 3 establirem dos tipus de resultats. En primer lloc, establirem les anomenades versions robustes dels teoremes clàssics de Kneser i Freiman. Robust aquí es refereix al fet que en comptes de demanar informació estructural sobre els conjunts constituents amb el coneixement que la seva suma és petita, només necessitem que això sigui vàlid per a un subconjunt gran passa si només volem donar una informació estructural per a gairebé tots els parells de conjunts amb una suma d'una mida determinada? Donem un teorema d'estructura aproximat que s'aplica a gairebé la majoria dels rangs possibles per la mida dels conjunts suma. Sistemes de conjunts de Sidon. Les preguntes clàssiques sobre els conjunts de Sidon són determinar la seva mida màxima o saber quan un conjunt aleatori és un conjunt de Sidon. Al capítol 4 generalitzem la noció de conjunts de Sidon per establir sistemes i establim els límits corresponents per a aquestes dues preguntes. També demostrem un resultat de densitat relativa, resultat condicionat a una conjectura sobre l'estructura específica dels sistemes màxims de Sidon. Conjunts independents en hipergrafs. El mètode dels contenidors d'hipergrafs és una eina general que es pot utilitzar per obtenir resultats sobre el nombre i l'estructura de conjunts independents en els hipergrafs. La connexió amb la combinatòria additiva apareix perquè molts problemes additius es poden codificar com l'estudi de conjunts independents en hipergrafs.Postprint (published version
When the sieve works
We are interested in classifying those sets of primes such that
when we sieve out the integers up to by the primes in we
are left with roughly the expected number of unsieved integers. In particular,
we obtain the first general results for sieving an interval of length with
primes including some in , using methods motivated by additive
combinatorics.Comment: 26 pages. Final version, published in Duke Math. J. Extended the
results of Section 2. Some other minor change
WHEN THE SIEVE WORKS
We are interested in classifying those sets of primes P such that when we sieve out the integers up to x by the primes in P-c we are left with roughly the expected number of unsieved integers. In particular, we obtain the first general results for sieving an interval of length x with primes including some in (root x, x], using methods motivated by additive combinatorics.</p
Sobolev, Besov and Triebel-Lizorkin spaces on quantum tori
This paper gives a systematic study of Sobolev, Besov and Triebel-Lizorkin
spaces on a noncommutative -torus (with a
skew symmetric real -matrix). These spaces share many properties
with their classical counterparts. We prove, among other basic properties, the
lifting theorem for all these spaces and a Poincar\'e type inequality for
Sobolev spaces. We also show that the Sobolev space
coincides with the Lipschitz space of order
, already studied by Weaver in the case . We establish the embedding
inequalities of all these spaces, including the Besov and Sobolev embedding
theorems. We obtain Littlewood-Paley type characterizations for Besov and
Triebel-Lizorkin spaces in a general way, as well as the concrete ones in terms
of the Poisson, heat semigroups and differences. Some of them are new even in
the commutative case, for instance, our Poisson semigroup characterizations
improve the classical ones. As a consequence of the characterization of the
Besov spaces by differences, we extend to the quantum setting the recent
results of Bourgain-Br\'ezis -Mironescu and Maz'ya-Shaposhnikova on the limits
of Besov norms. The same characterization implies that the Besov space
for is the quantum
analogue of the usual Zygmund class of order . We investigate the
interpolation of all these spaces, in particular, determine explicitly the
K-functional of the couple , which is the quantum analogue of a classical
result due to Johnen and Scherer. Finally, we show that the completely bounded
Fourier multipliers on all these spaces do not depend on the matrix ,
so coincide with those on the corresponding spaces on the usual -torus
Recommended from our members
Problems and results on linear hypergraphs
In this thesis, we tackle several problems involving the study of 3-uniform, linear hypergraphs satisfying some additional structural constraint.
We begin with a problem of Hrushovski concerning Latin squares satisfying a partial associativity condition. From an Latin square one can define a binary operation , and is associative if and only if is a group multiplication table. Hrushovski asked whether, if is only associative a positive proportion of the time, must still in some sense be close to a group multiplication table. This problem manifests a well-studied combinatorial theme, in which a local structural constraint is relaxed (first to a `99' version and then to a `1' version) and the global consequences of the relaxed constraints are analysed. We show that the partial associativity condition is sufficient to deduce powerful global information, allowing us to find within a large subset with group-like structure. Since Latin squares can be regarded as 3-uniform, linear hypergraphs, and the partial associativity condition can be formulated in terms of the count of a particular subhypergraph, we are able to apply purely combinatorial methods to a problem that touches algebra, model theory and geometric group theory.
We then take this problem further. A condition due to Thomsen provides a combinatorial constraint which, if satisfied by the Latin square , proves that is in fact the multiplication table of an abelian group. It is then natural to ask whether a relaxed version of this result is also attainable, and by extending our methods we are able to prove a result of this flavour. Since the combinatorial obstructions to commutativity of are far more complex than those for associativity, topological complications arise that are not present in the earlier work.
We also study a problem of Loh concerning sequences of triples of integers from satisfying a certain `increasing' property. Loh studied the maximum length of such a sequence, improving a trivial upper bound of to using the triangle removal lemma and conjecturing that a natural construction of length is best possible. We provide the first power-type improvement to the upper bound, showing that there exists such that the length is bounded by . By viewing the triples as edges in a 3-uniform hypergraph, the increasing property shows that the hypergraph is linear and provides further restrictions in terms of forbidden subhypergraphs. By considering this formulation, we provide links to various important open problems including the Brown--Erd\H os--S\'os conjecture.
Finally, we present a collection of shorter results. In work connecting to the earlier chapters, we resolve the Brown--Erd\H os--S\'os conjecture in the context of hypergraphs with a group structure, and show moreover that subsets of group multiplication tables exhibit local density far beyond what can be hoped for in general. In work less closely connected to the main theme of the thesis, we also answer a question of Leader, Mili\'cevi\'c and Tan concerning partitions of boxes, consider a problem on projective cubes in , and resolve a conjecture concerning a diffusion process on graphs