Discrete Analysis最新文献

{"title":"A short proof of a conjecture of Erd\"os proved by Moreira, Richter and Robertson","authors":"B. Host","doi":"10.19086/DA.11129","DOIUrl":"https://doi.org/10.19086/DA.11129","url":null,"abstract":"We give a short proof of a sumset conjecture of Erdos, recently proved by Moreira, Richter and Robertson: every subset of the integers of positive density contains the sum of two infinite sets. The proof is written in the framework of classical ergodic theory.","PeriodicalId":37312,"journal":{"name":"Discrete Analysis","volume":" ","pages":""},"PeriodicalIF":1.1,"publicationDate":"2019-04-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47198854","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}

{"title":"Failure of the trilinear operator space Grothendieck theorem","authors":"J. Briet, C. Palazuelos","doi":"10.19086/da.8805","DOIUrl":"https://doi.org/10.19086/da.8805","url":null,"abstract":"We give a counterexample to a trilinear version of the operator space Grothendieck theorem. In particular, we show that for trilinear forms on l∞, the ratio of the symmetrized completely bounded norm and the jointly completely bounded norm is in general unbounded. The proof is based on a non-commutative version of the generalized von Neumann inequality from additive combinatorics.","PeriodicalId":37312,"journal":{"name":"Discrete Analysis","volume":" ","pages":""},"PeriodicalIF":1.1,"publicationDate":"2018-12-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46628535","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}

{"title":"On an almost all version of the Balog-Szemeredi-Gowers theorem","authors":"X. Shao","doi":"10.19086/DA.9095","DOIUrl":"https://doi.org/10.19086/DA.9095","url":null,"abstract":"We deduce, as a consequence of the arithmetic removal lemma, an almost-all version of the Balog-Szemer'{e}di-Gowers theorem: For any $Kgeq 1$ and $varepsilon > 0$, there exists $delta = delta(K,varepsilon)>0$ such that the following statement holds: if $|A+_{Gamma}A| leq K|A|$ for some $Gamma geq (1-delta)|A|^2$, then there is a subset $A' subset A$ with $|A'| geq (1-varepsilon)|A|$ such that $|A'+A'| leq |A+_{Gamma}A| + varepsilon |A|$. We also discuss issues around quantitative bounds in this statement, in particular showing that when $A subset mathbb{Z}$ the dependence of $delta$ on $epsilon$ cannot be polynomial for any fixed $K>2$.","PeriodicalId":37312,"journal":{"name":"Discrete Analysis","volume":"1 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2018-11-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"68393490","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}

{"title":"On the Lehmer conjecture and counting in finite fields","authors":"E. Breuillard, P. P. Varj'u","doi":"10.19086/da.8306","DOIUrl":"https://doi.org/10.19086/da.8306","url":null,"abstract":"We give a reformulation of the Lehmer conjecture about algebraic integers in terms of a simple counting problem modulo p.","PeriodicalId":37312,"journal":{"name":"Discrete Analysis","volume":" ","pages":""},"PeriodicalIF":1.1,"publicationDate":"2018-10-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46616963","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}

{"title":"Logarithmic bounds for Roth's theorem via almost-periodicity","authors":"T. Bloom, Olof Sisask","doi":"10.19086/da.7884","DOIUrl":"https://doi.org/10.19086/da.7884","url":null,"abstract":"Logarithmic bounds for Roth's theorem via almost-periodicity, Discrete Analysis 2019:4, 20pp.\u0000\u0000A central result of additive combinatorics, Roth's theorem, asserts that for every $delta>0$ there exists $N$ such that every subset of ${1,2,dots,N}$ of size at least $delta N$ contains an arithmetic progression of length 3. This is the first non-trivial case of Szemeredi's theorem, proved over 20 years later, which is the corresponding statement for progressions of general length. If we define $rho_k(N)$ to be the minimum real number such that every subset of ${1,2,dots,N}$ of density at least $rho_k(N)$ contains an arithmetic progression of length $k$, then Roth's theorem asserts that $rho_3(N)to 0$ and Szemeredi's theorem asserts that $rho_k(N)to 0$ for all $k$. \u0000\u0000These results leave open the question of bounds. Roth's proof shows that $rho_3(N)leq C/loglog N$ for an absolute constant $C$. Following a sequence of improvements by Szemeredi, Heath-Brown, Bourgain and Sanders, the current record, due to the first author of this paper, stands at $C(loglog N)^4/log N$. This is tantalizingly close to $1/log N$, which is an important barrier because if one could get past it then one would be able to prove that every set $Asubsetmathbb N$ such that $sum_{xin A}x^{-1}=infty$ contains an arithmetic progression of length 3, which is the first non-trivial case of perhaps the most famous of all conjectures of Erdős.\u0000\u0000At the time of writing, the problem of beating the log barrier is particularly alive, because there is some evidence that we already have the technology needed to solve it. This evidence comes from a closely related problem, the cap-set problem, which concerns the density that a subset $Asubsetmathbb F_3^n$ must have in order to contain an affine line, which is the natural notion of an arithmetic progression of length 3 in $mathbb F_3^n$. For a long time the best known upper bound was stuck at $Cn^{-1}$, which is also a logarithmic bound, since the cardinality of $mathbb F_3^n$ is $3^n$. Then a few years ago, Michael Bateman and Nets Katz improved the bound to $Cn^{-(1+epsilon)}$ for a small positive $epsilon$, and more recently, in a spectacular development, Jordan Ellenberg and Dion Gijswijt, building on work of Ernie Croot, Seva Lev and Peter Pach, obtained an upper bound of $c^n$ for a constant $c<1$. Ellenberg and Gijswijt used the polynomial method, and it is far from clear whether any analogue of that method can be made to work for Roth's theorem, so there is continued interest in the argument of Bateman and Katz, which involved a delicate analysis of the structure of the set of large Fourier coefficients of a dense set. Could a similar analysis be used to improve the current record for Roth's theorem by a power of $log n$? There are significant difficulties (not least of which is the complexity of the arguments that one would be trying to combine), but there does not appear to be a clear reason to suppose that such a programme cannot be carr","PeriodicalId":37312,"journal":{"name":"Discrete Analysis","volume":" ","pages":""},"PeriodicalIF":1.1,"publicationDate":"2018-10-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43914564","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}

{"title":"Approximate lattices and Meyer sets in nilpotent Lie groups","authors":"S. Machado","doi":"10.19086/da.11886","DOIUrl":"https://doi.org/10.19086/da.11886","url":null,"abstract":"We show that uniform approximate lattices in nilpotent Lie groups are subsets of model sets. This extends Y.Meyer's theorem about quasi-crystals in Euclidean spaces. We derive from this structure theorem a characterisation of connected, simply connected, nilpotent Lie groups containing approximate lattices as the groups whose Lie algebra have structure constants lying in $overline{mathbb{Q}}$.","PeriodicalId":37312,"journal":{"name":"Discrete Analysis","volume":" ","pages":""},"PeriodicalIF":1.1,"publicationDate":"2018-10-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47802879","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}

{"title":"Finitely forcible graphons with an almost arbitrary structure","authors":"D. Král, L. Lov'asz, Jonathan A. Noel, Jakub Sosnovec","doi":"10.19086/DA.12058","DOIUrl":"https://doi.org/10.19086/DA.12058","url":null,"abstract":"A basic result from the theory of quasirandom graphs, due to Andrew Thomason, is that if is a graph with vertices and density , and if the number of 4-cycles in is approximately , then resembles a random graph of the same density. In particular, between any two sets and of vertices the number of edges is approximately . (Here, “approximately” means \"to within a small fraction of , so the statement is non-trivial only for sets and that are not too small.)","PeriodicalId":37312,"journal":{"name":"Discrete Analysis","volume":" ","pages":""},"PeriodicalIF":1.1,"publicationDate":"2018-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44172926","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}

{"title":"On sets defining few ordinary hyperplanes","authors":"Aaron Lin, K. Swanepoel","doi":"10.19086/da.11949","DOIUrl":"https://doi.org/10.19086/da.11949","url":null,"abstract":"Let P be a set of n points in real projective d-space, not all contained in a hyperplane, such that any d points span a hyperplane. An ordinary hyperplane of P is a hyperplane containing exactly d points of P. We show that if d > 4, the number of ordinary hyperplanes of P is at least 􀀀n􀀀1 d􀀀1 􀀀 Od(nb(d􀀀1)=2c) if n is suciently large depending on d. This bound is tight, and given d, we can calculate the exact minimum number for suciently large n. This is a consequence of a structure theorem for sets with few ordinary hyperplanes: For any d > 4 and K > 0, if n > CdK8 for some constant Cd > 0 depending on d, and P spans at most K 􀀀n􀀀1 d􀀀1 ordinary hyperplanes, then all but at most Od(K) points of P lie on a hyperplane, an elliptic normal curve, or a rational acnodal curve. We also nd the maximum number of (d+1)-point hyperplanes, solving a d-dimensional analogue of the orchard problem. Our proofs rely on Green and Tao's results on ordinary lines, our earlier work on the 3-dimensional case, as well as results from classical algebraic geometry.","PeriodicalId":37312,"journal":{"name":"Discrete Analysis","volume":" ","pages":""},"PeriodicalIF":1.1,"publicationDate":"2018-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45284517","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}

{"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":" ","pages":""},"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}

{"title":"Stronger arithmetic equivalence","authors":"Shachar Lovett","doi":"10.19086/da.8654","DOIUrl":"https://doi.org/10.19086/da.8654","url":null,"abstract":"This paper considers two alternative strengthenings of the notion of arithmetic equivalence, which the author calls local integral equivalence and solvable equivalence. (The latter turns out to be strictly stronger than the former.) They have the advantage of being easier to check than Prasad’s notion, which the author calls integral equivalence. Furthermore, solvable equivalence, which the author shows does not imply integral equivalence, is nevertheless a sufficient condition to imply that the invariants considered by Prasad are equal. This opens the door to easier proofs of Prasad’s result, and lessens the reliance on Scott’s construction: the author finds a generalization of this construction that yields infinitely many examples of solvable equivalence. The paper also contains several examples to clarify the relationships between the various different notions of equivalence. Some of these examples (which are mainly found with the help of a computer) answer open questions from the group theory literature.","PeriodicalId":37312,"journal":{"name":"Discrete Analysis","volume":" ","pages":""},"PeriodicalIF":1.1,"publicationDate":"2018-06-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47120580","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}