{"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
{"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}
{"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}
{"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}
{"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}
{"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}
{"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
{"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}
{"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}
{"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}
求助内容：
应助结果提醒方式：
京公网安备 11010802042870号
