Motivated by a recent result of Prasad, we consider three stronger notions of arithmetic equivalence: local integral equivalence, integral equivalence, and solvable equivalence. In addition to having the same Dedekind zeta function (the usual notion of arithmetic equivalence), number fields that are equivalent in any of these stronger senses must have the same class number, and solvable equivalence forces an isomorphism of adele rings. Until recently the only nontrivial example of integral and solvable equivalence arose from a group-theoretic construction of Scott that was exploited by Prasad. Here we provide infinitely many distinct examples of solvable equivalence, including a family that contains Scott's construction as well as an explicit example of degree 96. We also construct examples that address questions of Scott, and of Guralnick and Weiss, and shed some light on a question of Prasad.
{"title":"Stronger arithmetic equivalence","authors":"Andrew Sutherland","doi":"10.19086/da.29452","DOIUrl":"https://doi.org/10.19086/da.29452","url":null,"abstract":"Motivated by a recent result of Prasad, we consider three stronger notions of arithmetic equivalence: local integral equivalence, integral equivalence, and solvable equivalence. In addition to having the same Dedekind zeta function (the usual notion of arithmetic equivalence), number fields that are equivalent in any of these stronger senses must have the same class number, and solvable equivalence forces an isomorphism of adele rings. Until recently the only nontrivial example of integral and solvable equivalence arose from a group-theoretic construction of Scott that was exploited by Prasad. Here we provide infinitely many distinct examples of solvable equivalence, including a family that contains Scott's construction as well as an explicit example of degree 96. We also construct examples that address questions of Scott, and of Guralnick and Weiss, and shed some light on a question of Prasad.","PeriodicalId":37312,"journal":{"name":"Discrete Analysis","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2021-04-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45240803","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}
Suppose we choose $N$ points uniformly randomly from a convex body in $d$ dimensions. How large must $N$ be, asymptotically with respect to $d$, so that the convex hull of the points is nearly as large as the convex body itself? It was shown by Dyer-Furedi-McDiarmid that exponentially many samples suffice when the convex body is the hypercube, and by Pivovarov that the Euclidean ball demands roughly $d^{d/2}$ samples. We show that when the convex body is the simplex, exponentially many samples suffice; this then implies the same result for any convex simplicial polytope with at most exponentially many faces.
{"title":"Random volumes in d-dimensional polytopes","authors":"A. Frieze, W. Pegden, T. Tkocz","doi":"10.19086/DA.17109","DOIUrl":"https://doi.org/10.19086/DA.17109","url":null,"abstract":"Suppose we choose $N$ points uniformly randomly from a convex body in $d$ dimensions. How large must $N$ be, asymptotically with respect to $d$, so that the convex hull of the points is nearly as large as the convex body itself? It was shown by Dyer-Furedi-McDiarmid that exponentially many samples suffice when the convex body is the hypercube, and by Pivovarov that the Euclidean ball demands roughly $d^{d/2}$ samples. We show that when the convex body is the simplex, exponentially many samples suffice; this then implies the same result for any convex simplicial polytope with at most exponentially many faces.","PeriodicalId":37312,"journal":{"name":"Discrete Analysis","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2020-02-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43115935","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}
Let $mathcal{L}$ be a family of lines and let $mathcal{P}$ be a family of $k$-planes in $mathbb{F}^n$ where $mathbb{F}$ is a field. In our first result we show that the number of joints formed by a $k$-plane in $mathcal{P}$ together with $(n-k)$ lines in $mathcal{L}$ is $O_n(|mathcal{L}||mathcal{P}|^{1/(n-k)}$). This is the first sharp result for joints involving higher-dimensional affine subspaces, and it holds in the setting of arbitrary fields $mathbb{F}$. In contrast, for our second result, we work in the three-dimensional Euclidean space $mathbb{R}^3$, and we establish the Kakeya-type estimate begin{equation*}sum_{x in J} left(sum_{ell in mathcal{L}} chi_ell(x)right)^{3/2} lesssim |mathcal{L}|^{3/2}end{equation*} where $J$ is the set of joints formed by $mathcal{L}$; such an estimate fails in the setting of arbitrary fields. This result strengthens the known estimates for joints, including those counting multiplicities. Additionally, our techniques yield significant structural information on quasi-extremisers for this inequality.
设$mathcal｛L｝$是一个线族，设$math cal｛P｝$为$mathbb｛F｝^n$中$k$平面的族，其中$mathbb{F｝$是字段。在我们的第一个结果中，我们证明了$mathcal｛P｝$中的$k$平面与$mathical｛L｝$的$（n-k）$线形成的关节数为$O_n（|mathcal{L｝|mathcal{P｝|^｛1/（n-k）｝$）。这是涉及高维仿射子空间的关节的第一个尖锐结果，并且它在任意域$mathbb｛F｝$的设置中成立。相反，对于我们的第二个结果，我们在三维欧几里得空间$mathbb｛R｝^3$中工作，并且我们建立了Kakeya型估计 begin｛equipment*｝sum_｛x In J｝left（sum_{ellInmathcal｛L｝｝chi_ell（x）right）^｛3/2｝lesssim|mathcal{L｝|^｛3/3｝end｛equation*｝，其中$J$是由$mathcal｝$形成的关节集；这样的估计在任意字段的设置中失败。这一结果加强了已知的节理估计，包括那些计算乘数的估计。此外，我们的技术产生了关于这个不等式的拟极值的重要结构信息。
{"title":"Joints formed by lines and a $k$-plane, and a discrete estimate of Kakeya type","authors":"A. Carbery, Marina Iliopoulou","doi":"10.19086/DA.18361","DOIUrl":"https://doi.org/10.19086/DA.18361","url":null,"abstract":"Let $mathcal{L}$ be a family of lines and let $mathcal{P}$ be a family of $k$-planes in $mathbb{F}^n$ where $mathbb{F}$ is a field. In our first result we show that the number of joints formed by a $k$-plane in $mathcal{P}$ together with $(n-k)$ lines in $mathcal{L}$ is $O_n(|mathcal{L}||mathcal{P}|^{1/(n-k)}$). This is the first sharp result for joints involving higher-dimensional affine subspaces, and it holds in the setting of arbitrary fields $mathbb{F}$. In contrast, for our second result, we work in the three-dimensional Euclidean space $mathbb{R}^3$, and we establish the Kakeya-type estimate begin{equation*}sum_{x in J} left(sum_{ell in mathcal{L}} chi_ell(x)right)^{3/2} lesssim |mathcal{L}|^{3/2}end{equation*} where $J$ is the set of joints formed by $mathcal{L}$; such an estimate fails in the setting of arbitrary fields. This result strengthens the known estimates for joints, including those counting multiplicities. Additionally, our techniques yield significant structural information on quasi-extremisers for this inequality.","PeriodicalId":37312,"journal":{"name":"Discrete Analysis","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2019-11-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43994732","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}
We prove a new, efficient version of the hypergraph container theorems that is suited for hypergraphs with large uniformities. The main novelty is a refined approach to constructing containers that employs simple ideas from high-dimensional convex geometry. The existence of smaller families of containers for independent sets in such hypergraphs, which is guaranteed by the new theorem, allows us to improve upon the best currently known bounds for several problems in extremal graph theory, discrete geometry, and Ramsey theory.
{"title":"An efficient container lemma","authors":"J. Balogh, W. Samotij","doi":"10.19086/DA.17354","DOIUrl":"https://doi.org/10.19086/DA.17354","url":null,"abstract":"We prove a new, efficient version of the hypergraph container theorems that is suited for hypergraphs with large uniformities. The main novelty is a refined approach to constructing containers that employs simple ideas from high-dimensional convex geometry. The existence of smaller families of containers for independent sets in such hypergraphs, which is guaranteed by the new theorem, allows us to improve upon the best currently known bounds for several problems in extremal graph theory, discrete geometry, and Ramsey theory.","PeriodicalId":37312,"journal":{"name":"Discrete Analysis","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2019-10-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41742931","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}
Let $f in mathbb{R}[x]$ be a polynomial with real coefficients. We say that $f$ is eventually non-negative if $f^m$ has non-negative coefficients for all sufficiently large $m in mathbb{N}$. In this short note, we give a classification of all eventually non-negative polynomials. This generalizes a theorem of De Angelis, and proves a conjecture of Bergweiler, Eremenko and Sokal
设$f in mathbb{R}[x]$是一个实系数多项式。如果$f^m$对于所有足够大的$m 在mathbb{N}$中都有非负系数，我们说$f$最终是非负的。在这个简短的笔记中，我们给出了所有最终非负多项式的分类。推广了De Angelis的一个定理，证明了Bergweiler、Eremenko和Sokal的一个猜想
{"title":"A characterization of polynomials whose high powers have non-negative coefficients","authors":"Marcus Michelen, J. Sahasrabudhe","doi":"10.19086/DA.18560","DOIUrl":"https://doi.org/10.19086/DA.18560","url":null,"abstract":"Let $f in mathbb{R}[x]$ be a polynomial with real coefficients. We say that $f$ is eventually non-negative if $f^m$ has non-negative coefficients for all sufficiently large $m in mathbb{N}$. In this short note, we give a classification of all eventually non-negative polynomials. This generalizes a theorem of De Angelis, and proves a conjecture of Bergweiler, Eremenko and Sokal","PeriodicalId":37312,"journal":{"name":"Discrete Analysis","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2019-10-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48029879","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}
The Landau-Selberg-Delange method provides an asymptotic formula for the partial sums of a multiplicative function whose average value on primes is a fixed complex number $v$. The shape of this asymptotic implies that $f$ can get very small on average only if $v=0,-1,-2,dots$. Moreover, if $v<0$, then the Dirichlet series associated to $f$ must have a zero of multiplicity $-v$ at $s=1$. In this paper, we prove a converse result that shows that if $f$ is a multiplicative function that is bounded by a suitable divisor function, and $f$ has very small partial sums, then there must be finitely many real numbers $gamma_1$, $dots$, $gamma_m$ such that $f(p)approx -p^{igamma_1}-cdots-p^{-igamma_m}$ on average. The numbers $gamma_j$ correspond to ordinates of zeroes of the Dirichlet series associated to $f$, counted with multiplicity. This generalizes a result of the first author, who handled the case when $|f|le 1$ in previous work.
{"title":"The structure of multiplicative functions with small partial sums","authors":"Dimitris Koukoulopoulos, K. Soundararajan","doi":"10.19086/da.11963","DOIUrl":"https://doi.org/10.19086/da.11963","url":null,"abstract":"The Landau-Selberg-Delange method provides an asymptotic formula for the partial sums of a multiplicative function whose average value on primes is a fixed complex number $v$. The shape of this asymptotic implies that $f$ can get very small on average only if $v=0,-1,-2,dots$. Moreover, if $v<0$, then the Dirichlet series associated to $f$ must have a zero of multiplicity $-v$ at $s=1$. In this paper, we prove a converse result that shows that if $f$ is a multiplicative function that is bounded by a suitable divisor function, and $f$ has very small partial sums, then there must be finitely many real numbers $gamma_1$, $dots$, $gamma_m$ such that $f(p)approx -p^{igamma_1}-cdots-p^{-igamma_m}$ on average. The numbers $gamma_j$ correspond to ordinates of zeroes of the Dirichlet series associated to $f$, counted with multiplicity. This generalizes a result of the first author, who handled the case when $|f|le 1$ in previous work.","PeriodicalId":37312,"journal":{"name":"Discrete Analysis","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2019-09-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47888538","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}
The main result of this paper is a decomposition theorem for a measure on the one-dimensional torus. Given a sufficiently large subset $S$ of the positive integers, an arbitrary measure on the torus is decomposed as the sum of two measures. The first one $mu_1$ has the property that the random walk with initial distribution $mu_1$ evolved by the action of $S$ equidistributes very fast. The second measure $mu_2$ in the decomposition is concentrated on very small neighborhoods of a small number of points.
{"title":"Decomposition of random walk measures on the one-dimensional torus.","authors":"T. Gilat","doi":"10.19086/da.11888","DOIUrl":"https://doi.org/10.19086/da.11888","url":null,"abstract":"The main result of this paper is a decomposition theorem for a measure on the one-dimensional torus. Given a sufficiently large subset $S$ of the positive integers, an arbitrary measure on the torus is decomposed as the sum of two measures. The first one $mu_1$ has the property that the random walk with initial distribution $mu_1$ evolved by the action of $S$ equidistributes very fast. The second measure $mu_2$ in the decomposition is concentrated on very small neighborhoods of a small number of points.","PeriodicalId":37312,"journal":{"name":"Discrete Analysis","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2019-09-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41901623","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}
Consider a quadratic polynomial $fleft(xi_{1},dots,xi_{n}right)$ of independent Bernoulli random variables. What can be said about the concentration of $f$ on any single value? This generalises the classical Littlewood--Offord problem, which asks the same question for linear polynomials. As in the linear case, it is known that the point probabilities of $f$ can be as large as about $1/sqrt{n}$, but still poorly understood is the "inverse" question of characterising the algebraic and arithmetic features $f$ must have if it has point probabilities comparable to this bound. In this paper we prove some results of an algebraic flavour, showing that if $f$ has point probabilities much larger than $1/n$ then it must be close to a quadratic form with low rank. We also give an application to Ramsey graphs, asymptotically answering a question of Kwan, Sudakov and Tran.
{"title":"An algebraic inverse theorem for the quadratic Littlewood-Offord problem, and an application to Ramsey graphs","authors":"Matthew Kwan, Lisa Sauermann","doi":"10.19086/DA.14351","DOIUrl":"https://doi.org/10.19086/DA.14351","url":null,"abstract":"Consider a quadratic polynomial $fleft(xi_{1},dots,xi_{n}right)$ of independent Bernoulli random variables. What can be said about the concentration of $f$ on any single value? This generalises the classical Littlewood--Offord problem, which asks the same question for linear polynomials. As in the linear case, it is known that the point probabilities of $f$ can be as large as about $1/sqrt{n}$, but still poorly understood is the \"inverse\" question of characterising the algebraic and arithmetic features $f$ must have if it has point probabilities comparable to this bound. In this paper we prove some results of an algebraic flavour, showing that if $f$ has point probabilities much larger than $1/n$ then it must be close to a quadratic form with low rank. We also give an application to Ramsey graphs, asymptotically answering a question of Kwan, Sudakov and Tran.","PeriodicalId":37312,"journal":{"name":"Discrete Analysis","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2019-09-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44404767","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}
Jaehoon Kim, Hong Liu, O. Pikhurko, M. Sharifzadeh
The famous Erdős-Rademacher problem asks for the smallest number of $r$-cliques in a graph with the given number of vertices and edges. Despite decades of active attempts, the asymptotic value of this extremal function for all $r$ was determined only recently, by Reiher [Annals of Mathematics, 184 (2016) 683--707]. Here we describe the asymptotic structure of all almost extremal graphs. This task for $r=3$ was previously accomplished by Pikhurko and Razborov [Combinatorics, Probability and Computing, 26 (2017) 138--160].
著名的Erdõs-Rademacher问题要求在给定顶点和边数的图中存在最小数量的$r$-群。尽管进行了几十年的积极尝试，但直到最近，Reiher才确定了所有$r$的这个极值函数的渐近值[数学年鉴，184（2016）683-707]。这里我们描述了所有几乎极值图的渐近结构。Pikhurko和Razborov之前完成了$r=3$的这项任务[Combinatorics，Probability and Computing，26（2017）138-160]。
{"title":"Asymptotic Structure for the Clique Density Theorem","authors":"Jaehoon Kim, Hong Liu, O. Pikhurko, M. Sharifzadeh","doi":"10.19086/DA.18559","DOIUrl":"https://doi.org/10.19086/DA.18559","url":null,"abstract":"The famous Erdős-Rademacher problem asks for the smallest number of $r$-cliques in a graph with the given number of vertices and edges. Despite decades of active attempts, the asymptotic value of this extremal function for all $r$ was determined only recently, by Reiher [Annals of Mathematics, 184 (2016) 683--707]. Here we describe the asymptotic structure of all almost extremal graphs. This task for $r=3$ was previously accomplished by Pikhurko and Razborov [Combinatorics, Probability and Computing, 26 (2017) 138--160].","PeriodicalId":37312,"journal":{"name":"Discrete Analysis","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2019-06-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47408428","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}
We build a faithful action of Higman's group on the line by homeomorphisms, answering a question of Yves de Cornulier. As a by-product we obtain many quasimorphisms from the Higman group into the reals. We also show that every action by $C^1$-diffeomorphisms of Higman's group on the line or the circle is trivial.
我们通过同胚建立了Higman群在线上的忠实行动，回答了Yves de Cornulier的一个问题。作为副产品，我们得到了许多从Higman群到实数的拟态射。我们还证明了Higman群的$C^1$-微分同胚在直线或圆上的每一个作用都是平凡的。
{"title":"One-dimensional actions of Higman's group.","authors":"C. Rivas, Michele Triestino","doi":"10.19086/DA.11151","DOIUrl":"https://doi.org/10.19086/DA.11151","url":null,"abstract":"We build a faithful action of Higman's group on the line by homeomorphisms, answering a question of Yves de Cornulier. As a by-product we obtain many quasimorphisms from the Higman group into the reals. We also show that every action by $C^1$-diffeomorphisms of Higman's group on the line or the circle is trivial.","PeriodicalId":37312,"journal":{"name":"Discrete Analysis","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2019-05-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43500870","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}
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