Discrete Analysis最新文献

{"title":"Semicontinuity of structure for small sumsets in compact abelian groups","authors":"John T. Griesmer","doi":"10.19086/da.11089","DOIUrl":"https://doi.org/10.19086/da.11089","url":null,"abstract":"Semicontinuity of structure for small sumsets in compact abelian groups, Discrete Analysis 2019:18, 46 pp.\u0000\u0000The well-known Cauchy-Davenport theorem asserts that if $A$ and $B$ are two subsets of a cyclic group of prime order $p$, then $|A+B|geqmin{|A|+|B|-1,p}$. It was generalized to a suitable statement about finite subsets of arbitrary Abelian groups by Martin Kneser: note that if $A$ and $B$ are unions of cosets of a subgroup $H$, then $|A+B|$ can be as small as $|A|+|B|-|H|$, and Kneser's theorem takes account of this. The question of what one can say when the inequalities are sharp was answered by Kemperman, who provided a rather complicated structural characterization.\u0000\u0000One can ask a corresponding question when $G$ is a compact Hausdorff Abelian topological group with Haar measure $m$. Now we let $A$ and $B$ be $m$-measurable subsets such that $$m_*(A+B)leq m(A)+m(B).$$\u0000 \u0000Here $m_*$ is the inner $m$-measure, since $A+B$ does not have to be measurable. (Indeed, Sierpinski showed that there are two measure-zero sets $A,B$ of reals such that $A+B$ is not measurable.) \u0000\u0000Such pairs of sets were characterized by Kneser under the additional assumption that $G$ is connected. An obvious example is where $A$ and $B$ are subintervals of the circle group, and Kneser showed that, roughly speaking, every example is an inverse image of such an example under a surjective $m$-measurable homomorphism. When $G$ is disconnected the characterization of pairs satisfying $m_*(A+B)=m(A)+m(B)$ is more complicated. Building on work of Hamidoune, Rodseth, Serra, and Zemor, Grynkiewicz provided a complete characterization of such pairs for discrete abelian groups $G$. The author of this paper combined Grynkiewicz's and Kneser's proofs to extend this to arbitrary compact Hausdorff abelian groups.\u0000\u0000The aim of this paper is to prove a stability version of preceding results: this is the meaning of the phrase \"semicontinuity of structure\" in the title. In other words, the paper is concerned with what happens if $m_*(A+B)leq m(A)+m(B)+delta$ when $delta$ is sufficiently small as a function of $m(A)$ and $m(B)$. One of the main results is the following, which has a similar flavour to the triangle removal lemma. Define $A+_delta B$ to be the set of all \"$delta$-popular\" elements of $A+B$ -- that is, the set of all $xin G$ such that $m{ain A: x-ain B}geqdelta$. The author shows that for every $epsilon>0$ there exists $delta>0$ such that if $m(A+_delta B)leq m(A)+m(B)+delta$, then there exist approximations $A'$ and $B'$ such that $m(Atriangle A')+m(Btriangle B')<epsilon$ and $m(A'+B')leq m(A')+m(B')$. Since, as was mentioned above, pairs $A',B'$ with this stronger property have been classified, this gives a complete characterization of pairs $A,B$ with the weaker property. \u0000\u0000The proof of this result leads to the desired classification of pairs $A,B$ for which $m_*(A+B)leq m(A)+m(B)+delta$. No explicit dependence of $delta$ on $m(A)$ and $m(B)$ is provided, because ultraproduct","PeriodicalId":37312,"journal":{"name":"Discrete Analysis","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2018-07-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43338183","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}