29 research outputs found
Small asymmetric sumsets in elementary abelian 2-groups
Let A and B be subsets of an elementary abelian 2-group G, none of which are
contained in a coset of a proper subgroup. Extending onto potentially distinct
summands a result of Hennecart and Plagne, we show that if |A+B|<|A|+|B|, then
either A+B=G, or the complement of A+B in G is contained in a coset of a
subgroup of index at least 8, whence |A+B| is at least 7/8 |G|. We indicate
conditions for the containment to be strict, and establish a refinement in the
case where the sizes of A and B differ significantly.Comment: 6 page
Roth's theorem for four variables and additive structures in sums of sparse sets
We show that if a subset A of {1,...,N} does not contain any solutions to the
equation x+y+z=3w with the variables not all equal, then A has size at most
exp(-c(log N)^{1/7}) N, where c > 0 is some absolute constant. In view of
Behrend's construction, this bound is of the right shape: the exponent 1/7
cannot be replaced by any constant larger than 1/2.
We also establish a related result, which says that sumsets A+A+A contain
long arithmetic progressions if A is a subset of {1,...,N}, or high-dimensional
subspaces if A is a subset of a vector space over a finite field, even if A has
density of the shape above.Comment: 23 page
Entropy and set cardinality inequalities for partition-determined functions
A new notion of partition-determined functions is introduced, and several
basic inequalities are developed for the entropy of such functions of
independent random variables, as well as for cardinalities of compound sets
obtained using these functions. Here a compound set means a set obtained by
varying each argument of a function of several variables over a set associated
with that argument, where all the sets are subsets of an appropriate algebraic
structure so that the function is well defined. On the one hand, the entropy
inequalities developed for partition-determined functions imply entropic
analogues of general inequalities of Pl\"unnecke-Ruzsa type. On the other hand,
the cardinality inequalities developed for compound sets imply several
inequalities for sumsets, including for instance a generalization of
inequalities proved by Gyarmati, Matolcsi and Ruzsa (2010). We also provide
partial progress towards a conjecture of Ruzsa (2007) for sumsets in nonabelian
groups. All proofs are elementary and rely on properly developing certain
information-theoretic inequalities.Comment: 26 pages. v2: Revised version incorporating referee feedback plus
inclusion of some additional corollaries and discussion. v3: Final version
with minor corrections. To appear in Random Structures and Algorithm
Additive Combinatorics: A Menu of Research Problems
This text contains over three hundred specific open questions on various
topics in additive combinatorics, each placed in context by reviewing all
relevant results. While the primary purpose is to provide an ample supply of
problems for student research, it is hopefully also useful for a wider
audience. It is the author's intention to keep the material current, thus all
feedback and updates are greatly appreciated.Comment: This August 2017 version incorporates feedback and updates from
several colleague
Energies and structure of additive sets
In the paper we prove that any sumset or difference set has large E_3 energy.
Also, we give a full description of families of sets having critical relations
between some kind of energies such as E_k, T_k and Gowers norms. In particular,
we give criteria for a set to be a 1) set of the form H+L, where H+H is small
and L has "random structure", 2) set equals a disjoint union of sets H_j, each
H_j has small doubling, 3) set having large subset A' with 2A' is equal to a
set with small doubling and |A'+A'| \approx |A|^4 / \E(A).Comment: 52 page
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 tambĂ© anomenada teoria combinatòria de nombres” Ă©s la d’estudiar l'estructura additiva de conjunts en determinats grups ambient. La combinatòria extremal estudia quant de gran pot ser una col·lecciĂł d'objectes finits abans d'exhibir determinats requisits estructurals. La combinatòria probabilĂstica analitza estructures combinatòries aleatòries, identificant en particular l'estructura dels objectes combinatoris tĂpics. Entre els estudis mĂ©s celebrats hi ha el treball de grafs aleatoris iniciat per Erdös i RĂ©nyi. Un exemple especialment rellevant de com aquestes tres Ă rees s'entrellacen Ă©s el desenvolupament per Erdös del mètode probabilĂstic 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 intersecciĂł d'aquestes tres Ă rees, i apareixen en els problemes segĂĽents. 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 gairebĂ© sempre. La segĂĽent pregunta lògica Ă©s la segĂĽent. Suposem que estem en la zona en que hi haurĂ solucions enteres en el conjunt aleatori binomial, com es distribueixen aleshores aquestes solucions? Al capĂtol 1, avançarem per respondre aquesta pregunta proporcionant condicions suficients per a quan una gran varietat de solucions segueixen una distribuciĂł normal. TambĂ© parlarem de com, en determinats casos, aquestes condicions suficients tambĂ© sĂłn necessĂ ries. Conjunts amb suma acotada. Què es pot dir de l'estructura de dos conjunts finits en un grup abeliĂ si la seva suma de Minkowski no Ă©s molt mĂ©s gran que la dels conjunts? Un resultat clĂ ssic de Kneser diu que això pot passar si i nomĂ©s si la suma de Minkowski Ă©s periòdica respecte a un subgrup propi. En el capĂtol 3 establirem dos tipus de resultats. En primer lloc, establirem les anomenades versions robustes dels teoremes clĂ ssics de Kneser i Freiman. Robust aquĂ es refereix al fet que en comptes de demanar informaciĂł estructural sobre els conjunts constituents amb el coneixement que la seva suma Ă©s petita, nomĂ©s necessitem que això sigui vĂ lid per a un subconjunt gran passa si nomĂ©s volem donar una informaciĂł estructural per a gairebĂ© tots els parells de conjunts amb una suma d'una mida determinada? Donem un teorema d'estructura aproximat que s'aplica a gairebĂ© la majoria dels rangs possibles per la mida dels conjunts suma. Sistemes de conjunts de Sidon. Les preguntes clĂ ssiques sobre els conjunts de Sidon sĂłn determinar la seva mida mĂ xima o saber quan un conjunt aleatori Ă©s un conjunt de Sidon. Al capĂtol 4 generalitzem la nociĂł de conjunts de Sidon per establir sistemes i establim els lĂmits corresponents per a aquestes dues preguntes. TambĂ© demostrem un resultat de densitat relativa, resultat condicionat a una conjectura sobre l'estructura especĂfica dels sistemes mĂ xims de Sidon. Conjunts independents en hipergrafs. El mètode dels contenidors d'hipergrafs Ă©s una eina general que es pot utilitzar per obtenir resultats sobre el nombre i l'estructura de conjunts independents en els hipergrafs. La connexiĂł amb la combinatòria additiva apareix perquè molts problemes additius es poden codificar com l'estudi de conjunts independents en hipergrafs.Postprint (published version
Approximate groups and doubling metrics
We develop a version of Freiman's theorem for a class of non-abelian groups,
which includes finite nilpotent, supersolvable and solvable A-groups. To do
this we have to replace the small doubling hypothesis with a stronger relative
polynomial growth hypothesis akin to that in Gromov's theorem (although with an
effective range), and the structures we find are balls in (left and right)
translation invariant pseudo-metrics with certain well behaved growth
estimates.
Our work complements three other recent approaches to developing non-abelian
versions of Freiman's theorem by Breuillard and Green, Fischer, Katz and Peng,
and Tao.Comment: 21 pp. Corrected typos. Changed title from `From polynomial growth to
metric balls in monomial groups