The sum of the absolute values of the Fourier coefficients of a function $f:mathbb{F}_2^n to mathbb{R}$ is called the spectral norm of $f$. Green and Sanders' quantitative version of Cohen's idempotent theorem states that if the spectral norm of $f:mathbb{F}_2^n to {0,1}$ is at most $M$, then the support of $f$ belongs to the ring of sets generated by at most $ell(M)$ cosets, where $ell(M)$ is a constant that only depends on $M$. We prove that the above statement can be generalized to emph{approximate} spectral norms if and only if the support of $f$ and its complement satisfy a certain arithmetic connectivity condition. In particular, our theorem provides a new proof of the quantitative Cohen's theorem for $mathbb{F}_2^n$.
函数$f:mathbb{F}_2^n to mathbb{R}$的傅里叶系数绝对值之和称为$f$的谱规范。格林和桑德斯的科恩等价定理的定量版本指出，如果$f:mathbb{F}_2^n to {0,1}$的谱规范至多为$M$，那么$f$的支持属于至多由$ell(M)$余集生成的集合环，其中$ell(M)$是一个只取决于$M$的常数。我们证明，当且仅当 $f$ 的支持及其补集满足一定的算术连通性条件时，上述声明可以推广到 emph{approximate}spectral norms。特别是，我们的定理为 $mathbb{F}_2^n$ 的定量科恩定理提供了新的证明。
{"title":"Boolean Functions with Small Approximate Spectral Norm","authors":"Tsun-Ming Cheung, Hamed Hatami, Rosie Zhao, Itai Zilberstein","doi":"arxiv-2409.10634","DOIUrl":"https://doi.org/arxiv-2409.10634","url":null,"abstract":"The sum of the absolute values of the Fourier coefficients of a function\u0000$f:mathbb{F}_2^n to mathbb{R}$ is called the spectral norm of $f$. Green and\u0000Sanders' quantitative version of Cohen's idempotent theorem states that if the\u0000spectral norm of $f:mathbb{F}_2^n to {0,1}$ is at most $M$, then the\u0000support of $f$ belongs to the ring of sets generated by at most $ell(M)$\u0000cosets, where $ell(M)$ is a constant that only depends on $M$. We prove that the above statement can be generalized to emph{approximate}\u0000spectral norms if and only if the support of $f$ and its complement satisfy a\u0000certain arithmetic connectivity condition. In particular, our theorem provides\u0000a new proof of the quantitative Cohen's theorem for $mathbb{F}_2^n$.","PeriodicalId":501407,"journal":{"name":"arXiv - MATH - Combinatorics","volume":"30 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142255067","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Christian HerrmannTechnische Universität Darmstadt, Dale R. Worley
For many equation-theoretical questions about modular lattices, Hall and Dilworth give a useful construction: Let $L_0$ be a lattice with largest element $u_0$, $L_1$ be a lattice disjoint from $L_0$ with smallest element $v_1$, and $a in L_0$, $b in L_1$ such that the intervals $[a, u_0]$ and $[v_1, b]$ are isomorphic. Then, after identifying those intervals you obtain $L_0 cup L_1$, a lattice structure whose partial order is the transitive relation generated by the partial orders of $L_0$ and $L_1$. It is modular if $L_0$ and $L_1$ are modular. Since in this construction the index set ${0, 1}$ is essentially a chain, this work presents a method -- termed S-glued -- whereby a general family $L_x (x in S)$ of lattices can specify a lattice with the small-scale lattice structure determined by the $L_x$ and the large-scale structure determined by $S$. A crucial application is representing finite-length modular lattices using projective geometries.
{"title":"S-Glued sums of lattices","authors":"Christian HerrmannTechnische Universität Darmstadt, Dale R. Worley","doi":"arxiv-2409.10738","DOIUrl":"https://doi.org/arxiv-2409.10738","url":null,"abstract":"For many equation-theoretical questions about modular lattices, Hall and\u0000Dilworth give a useful construction: Let $L_0$ be a lattice with largest\u0000element $u_0$, $L_1$ be a lattice disjoint from $L_0$ with smallest element\u0000$v_1$, and $a in L_0$, $b in L_1$ such that the intervals $[a, u_0]$ and\u0000$[v_1, b]$ are isomorphic. Then, after identifying those intervals you obtain\u0000$L_0 cup L_1$, a lattice structure whose partial order is the transitive\u0000relation generated by the partial orders of $L_0$ and $L_1$. It is modular if\u0000$L_0$ and $L_1$ are modular. Since in this construction the index set ${0,\u00001}$ is essentially a chain, this work presents a method -- termed S-glued --\u0000whereby a general family $L_x (x in S)$ of lattices can specify a lattice\u0000with the small-scale lattice structure determined by the $L_x$ and the\u0000large-scale structure determined by $S$. A crucial application is representing\u0000finite-length modular lattices using projective geometries.","PeriodicalId":501407,"journal":{"name":"arXiv - MATH - Combinatorics","volume":"29 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142268997","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
A regular-graph design is a block design for which a pair ${a,b}$ of distinct points occurs in $lambda+1$ or $lambda$ blocks depending on whether ${a,b}$ is or is not an edge of a given $delta$-regular graph. Our paper describes a specific construction for regular-graph designs with $lambda = 1$ and block size $delta + 1$. We show that for $delta in {2,3}$, certain necessary conditions for the existence of such a design with $n$ points are sufficient, with two exceptions in each case and two possible exceptions when $delta = 3$. We also construct designs of orders 105 and 117 for connected 4-regular graphs.
{"title":"A construction for regular-graph designs","authors":"Anthony Forbes, Carrie Rutherford","doi":"arxiv-2409.10159","DOIUrl":"https://doi.org/arxiv-2409.10159","url":null,"abstract":"A regular-graph design is a block design for which a pair ${a,b}$ of\u0000distinct points occurs in $lambda+1$ or $lambda$ blocks depending on whether\u0000${a,b}$ is or is not an edge of a given $delta$-regular graph. Our paper\u0000describes a specific construction for regular-graph designs with $lambda = 1$\u0000and block size $delta + 1$. We show that for $delta in {2,3}$, certain\u0000necessary conditions for the existence of such a design with $n$ points are\u0000sufficient, with two exceptions in each case and two possible exceptions when\u0000$delta = 3$. We also construct designs of orders 105 and 117 for connected\u00004-regular graphs.","PeriodicalId":501407,"journal":{"name":"arXiv - MATH - Combinatorics","volume":"201 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142255072","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Let $mathcal{H}$ be a family of graphs. The generalized Tur'an number $ex(n, K_r, mathcal{H})$ is the maximum number of copies of the clique $K_r$ in any $n$-vertex $mathcal{H}$-free graph. In this paper, we determine the value of $ex(n, K_r, {P_k, K_m } )$ for sufficiently large $n$ with an exceptional case, and characterize all corresponding extremal graphs, which generalizes and strengthens the results of Katona and Xiao [EJC, 2024] on $ex(n, K_2, {P_k, K_m } )$. For the exceptional case, we obtain a tight upper bound for $ex(n, K_r, {P_k, K_m } )$ that confirms a conjecture on $ex(n, K_2, {P_k, K_m } )$ posed by Katona and Xiao.
{"title":"Generalized Turán problem for a path and a clique","authors":"Xiaona Fang, Xiutao Zhu, Yaojun Chen","doi":"arxiv-2409.10129","DOIUrl":"https://doi.org/arxiv-2409.10129","url":null,"abstract":"Let $mathcal{H}$ be a family of graphs. The generalized Tur'an number\u0000$ex(n, K_r, mathcal{H})$ is the maximum number of copies of the clique $K_r$\u0000in any $n$-vertex $mathcal{H}$-free graph. In this paper, we determine the\u0000value of $ex(n, K_r, {P_k, K_m } )$ for sufficiently large $n$ with an\u0000exceptional case, and characterize all corresponding extremal graphs, which\u0000generalizes and strengthens the results of Katona and Xiao [EJC, 2024] on\u0000$ex(n, K_2, {P_k, K_m } )$. For the exceptional case, we obtain a tight upper\u0000bound for $ex(n, K_r, {P_k, K_m } )$ that confirms a conjecture on $ex(n,\u0000K_2, {P_k, K_m } )$ posed by Katona and Xiao.","PeriodicalId":501407,"journal":{"name":"arXiv - MATH - Combinatorics","volume":"117 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142255074","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Let $mathbb{F}$ be a field, and consider the hypercube ${ 0, 1 }^{n}$ in $mathbb{F}^{n}$. Sziklai and Weiner (Journal of Combinatorial Theory, Series A 2022) showed that if a polynomial $P ( X_{1}, dots, X_{n} ) in mathbb{F}[ X_{1}, dots, X_{n}]$ vanishes on every point of the hypercube ${0,1}^{n}$ except those with at most $r$ many ones then the degree of the polynomial will be at least $n-r$. This is a generalization of Alon and F"uredi's fundamental result (European Journal of Combinatorics 1993) about polynomials vanishing on every point of the hypercube except at the origin (point with all zero coordinates). Sziklai and Weiner proved their interesting result using M"{o}bius inversion formula and the Zeilberger method for proving binomial equalities. In this short note, we show that a stronger version of Sziklai and Weiner's result can be derived directly from Alon and F"{u}redi's result.
{"title":"About almost covering subsets of the hypercube","authors":"Arijit Ghosh, Chandrima Kayal, Soumi Nandi","doi":"arxiv-2409.10573","DOIUrl":"https://doi.org/arxiv-2409.10573","url":null,"abstract":"Let $mathbb{F}$ be a field, and consider the hypercube ${ 0, 1 }^{n}$ in\u0000$mathbb{F}^{n}$. Sziklai and Weiner (Journal of Combinatorial Theory, Series A\u00002022) showed that if a polynomial $P ( X_{1}, dots, X_{n} ) in mathbb{F}[\u0000X_{1}, dots, X_{n}]$ vanishes on every point of the hypercube ${0,1}^{n}$\u0000except those with at most $r$ many ones then the degree of the polynomial will\u0000be at least $n-r$. This is a generalization of Alon and F\"uredi's fundamental\u0000result (European Journal of Combinatorics 1993) about polynomials vanishing on\u0000every point of the hypercube except at the origin (point with all zero\u0000coordinates). Sziklai and Weiner proved their interesting result using\u0000M\"{o}bius inversion formula and the Zeilberger method for proving binomial\u0000equalities. In this short note, we show that a stronger version of Sziklai and\u0000Weiner's result can be derived directly from Alon and F\"{u}redi's result.","PeriodicalId":501407,"journal":{"name":"arXiv - MATH - Combinatorics","volume":"48 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142254809","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we present a method for constructing point primitive block transitive $t$-designs invariant under finite groups. Furthermore, we demonstrate that every point and block primitive $G$-invariant design can be generated using this method. Additionally, we establish the theoretical possibility of identifying all block transitive $G$-invariant designs. However, in practice, the feasibility of enumerating all designs for larger groups may be limited by the computational complexity involved.
{"title":"On point and block primitive designs invariant under permutation groups","authors":"Amin Saeidi","doi":"arxiv-2409.09730","DOIUrl":"https://doi.org/arxiv-2409.09730","url":null,"abstract":"In this paper, we present a method for constructing point primitive block\u0000transitive $t$-designs invariant under finite groups. Furthermore, we\u0000demonstrate that every point and block primitive $G$-invariant design can be\u0000generated using this method. Additionally, we establish the theoretical possibility of identifying all\u0000block transitive $G$-invariant designs. However, in practice, the feasibility\u0000of enumerating all designs for larger groups may be limited by the\u0000computational complexity involved.","PeriodicalId":501407,"journal":{"name":"arXiv - MATH - Combinatorics","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142269004","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Let G be an acyclic digraph, and let a, b, c, d be vertices, where a, b are sources, c, d are sinks, and every other vertex has in-degree and out-degree at least two. In 1985, Thomassen showed that there do not exist disjoint directed paths from a to c and from b to d, if and only if G can be drawn in a closed disc with a, b, c, d drawn in the boundary in order. We give a shorter proof.
设 G 是一个非循环数图，设 a、b、c、d 为顶点，其中 a、b 为源顶点，c、d 为汇顶顶点，每个其他顶点的入度和出度至少为 2。1985 年，托马森（Thomassen）证明了当且仅当 G 可以画成一个封闭的圆盘，并在边界上依次画出 a、b、c、d 时，不存在从 a 到 c 和从 b 到 d 的互不相交的有向路径。我们给出一个更简短的证明。
{"title":"Thomassen's theorem on the two-linkage problem in acyclic digraphs: a shorter proof","authors":"Paul Seymour","doi":"arxiv-2409.09758","DOIUrl":"https://doi.org/arxiv-2409.09758","url":null,"abstract":"Let G be an acyclic digraph, and let a, b, c, d be vertices, where a, b are\u0000sources, c, d are sinks, and every other vertex has in-degree and out-degree at\u0000least two. In 1985, Thomassen showed that there do not exist disjoint directed\u0000paths from a to c and from b to d, if and only if G can be drawn in a closed\u0000disc with a, b, c, d drawn in the boundary in order. We give a shorter proof.","PeriodicalId":501407,"journal":{"name":"arXiv - MATH - Combinatorics","volume":"23 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142255073","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Aaron Autry, Slade Gunter, Christopher Housholder, Steven Senger
We study questions inspired by ErdH os' celebrated distance problems with dot products in lieu of distances, and for more than a single pair of points. In particular, we study point configurations present in large finite point sets in the plane that are described by weighted trees. We give new lower bounds on the number of distinct sets of dot products serving as weights for a given type of tree in any large finite point set. We also as demonstrate the existence of many repetitions of some special sets of dot products occurring in a given type of tree in different constructions, narrowing gap between the best known upper and lower bounds on these configurations.
我们研究的问题受到 ErdH os 著名的距离问题的启发，即用点积代替距离，而且不止一对点。我们给出了新的下限，即在任何大型有限点集中，作为给定类型树权重的不同点积集的数量。我们还证明了在不同构造中出现在给定树类型中的一些特殊点积集存在许多重复，从而缩小了这些配置的已知上下限之间的差距。
{"title":"Bounds on distinct and repeated dot product trees","authors":"Aaron Autry, Slade Gunter, Christopher Housholder, Steven Senger","doi":"arxiv-2409.09683","DOIUrl":"https://doi.org/arxiv-2409.09683","url":null,"abstract":"We study questions inspired by ErdH os' celebrated distance problems with\u0000dot products in lieu of distances, and for more than a single pair of points.\u0000In particular, we study point configurations present in large finite point sets\u0000in the plane that are described by weighted trees. We give new lower bounds on\u0000the number of distinct sets of dot products serving as weights for a given type\u0000of tree in any large finite point set. We also as demonstrate the existence of\u0000many repetitions of some special sets of dot products occurring in a given type\u0000of tree in different constructions, narrowing gap between the best known upper\u0000and lower bounds on these configurations.","PeriodicalId":501407,"journal":{"name":"arXiv - MATH - Combinatorics","volume":"9 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142255075","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we present five different formulas for both discrete and fractional iterations of an invertible power series $f$ utilizing a novel and unifying approach from umbral calculus. Established formulas are extended, and their proofs simplified, while new formulas are introduced. In particular, through the use of $q$-calculus identities, we eliminate the requirement for $f'(0)$ to equal $1$ and, consequently, the corresponding new expressions for the iterative logarithm are derived.
{"title":"Explicit Expressions for Iterates of Power Series","authors":"Beauduin Kei","doi":"arxiv-2409.09809","DOIUrl":"https://doi.org/arxiv-2409.09809","url":null,"abstract":"In this paper, we present five different formulas for both discrete and\u0000fractional iterations of an invertible power series $f$ utilizing a novel and\u0000unifying approach from umbral calculus. Established formulas are extended, and\u0000their proofs simplified, while new formulas are introduced. In particular,\u0000through the use of $q$-calculus identities, we eliminate the requirement for\u0000$f'(0)$ to equal $1$ and, consequently, the corresponding new expressions for\u0000the iterative logarithm are derived.","PeriodicalId":501407,"journal":{"name":"arXiv - MATH - Combinatorics","volume":"16 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142255071","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We give an exact formula for the number of distinct square patterns of a given size that occur in the Squiral tiling.
我们给出了一个精确的公式，即在斯奎尔贴图中出现的给定大小的不同正方形图案的数量。
{"title":"The Pattern Complexity of the Squiral Tiling","authors":"Johan Nilsson","doi":"arxiv-2409.09847","DOIUrl":"https://doi.org/arxiv-2409.09847","url":null,"abstract":"We give an exact formula for the number of distinct square patterns of a\u0000given size that occur in the Squiral tiling.","PeriodicalId":501407,"journal":{"name":"arXiv - MATH - Combinatorics","volume":"102 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142254800","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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