检索结果-内蒙古大学图书馆

Owings-like theorems for infinitely many colours or finite monochromatic sets

ANNALS OF PURE AND APPLIED LOGIC 2024年第10期175卷

作者： Fernandez-Breton, David J. Rosales, Eliseo Sarmiento Vera, German Inst Politecn Nacl Escuela Super Fis & Matemat Inst Politecn Nacl S-NEdificio 9 Mexico City 07738 Mexico

Inspired by Owings's problem, we investigate whether, for a given an Abelian group G and cardinal numbers rc, 9 , every colouring c : G -> theta yields a subset X C_ G with |X| X | = rc such that X + X is monochromatic. (Owings's problem asks this for G = Z , 9 = 2 and rc = k0;0;this is known to be false for the same G and rc but 9 = 3.) We completely settle the question for rc and 9 both finite (by obtaining sufficient and necessary conditions for a positive answer) and for rc and 9 both infinite (with a negative answer). Also, in the case where 9 is infinite but rc is finite, we obtain some sufficient conditions for a negative answer as well as an example with a positive answer. (c) 2024 Elsevier B.V. All rights are reserved, including those for text and data mining, AI training, and similar technologies.

关键词： Owings's problem Ramsey-type theorems Combinatorial set theory Abelian group additive combinatorics

来源：评论

学校读者我要写书评

暂无评论

On sumsets involving kth powers of finite fields

引用

DISCRETE MATHEMATICS 2024年第4期347卷

作者： Wu, Hai-Liang Wei, Ning-Liu Li, Yu-Bo Nanjing Univ Posts & Telecommun Sch Sci Nanjing 210023 Peoples R China

In this paper, we study some topics concerning the additive decompositions of the set D-k of all kth power residues modulo a prime p. For example, given a positive integer k >= 2, we prove that lim( x ->+infinity) B(x)/ pi(x) = 0, where pi(x) is the number of primes p <= x and B(x) denotes the cardinality of the set { p <= x : p equivalent to 1 (mod k);D-k has a non -trivial 2-additive decomposition}. (c) 2024 Elsevier B.V. All rights reserved.

关键词： Sumsets kth power residues additive combinatorics

来源：评论

学校读者我要写书评

暂无评论

Sets without k-term progressions can have many shorter progressions

引用

RANDOM STRUCTURES & ALGORITHMS 2021年第3期58卷 383-389页

作者： Fox, Jacob Pohoata, Cosmin Stanford Univ Dept Math Stanford CA 94305 USA CALTECH Dept Math Pasadena CA 91125 USA

Let f(s, k)(n) be the maximum possible number of s-term arithmetic progressions in a set of n integers which contains no k-term arithmetic progression. For all fixed integers k > s >= 3, we prove that f(s, k)(n) = n(2 - o(1)), which answers an old question of Erdos. In fact, we prove upper and lower bounds for f(s, k)(n) which show that its growth is closely related to the bounds in Szemeredi's theorem.

关键词： additive combinatorics arithmetic progressions probabilistic methods Szemeré di&apos s theorem

来源：评论

学校读者我要写书评

暂无评论

On permutations of {1, ..., n} of and related topics

引用

JOURNAL OF ALGEBRAIC combinatorics 2021年第3期54卷 893-912页

作者： Sun, Zhi-Wei Nanjing Univ Dept Math Nanjing 210093 Peoples R China

In this paper, we study combinatorial aspects of permutations of {1, ..., n} and related topics. In particular, we prove that there is a unique permutation pi of {1, ..., n} such that all the numbers k + (pi k) (k = 1, ..., n) are powers of two. We also show that n vertical bar per[i(j-1)](1 <= n, j <= n) for any integer n > 2. We conjecture that if a group G contains no element of order among 2, ..., n + 1 then any A subset of G with vertical bar A vertical bar = n can be written as {a1, ..., a(n)} with a(1), a(2)(2), ..., a(n)(n) pairwise distinct. This conjecture is confirmed when G is a torsion-free abelian group. We also prove that for any finite subset A of a torsion-free abelian group G with vertical bar A vertical bar = n > 3, there is a numbering a(1), ..., a(n) of all the elements of A such that all the n sums a(1) + a(2) + a(3), a(2 )+ a(3) + a(4), ..., a(n-2) + a(n-1) + a(n), a(n-1) + a(n) + a(1), a(n) + a(1) + a(2) are pairwise distinct, and conjecture that this remains valid if G is cyclic.

关键词： additive combinatorics Permutations Powers of two Permanents Groups

来源：评论

学校读者我要写书评

暂无评论

KLEINIAN SCHOTTKY GROUPS, PATTERSON-SULLIVAN MEASURES, AND FOURIER DECAY

引用

DUKE MATHEMATICAL JOURNAL 2021年第4期170卷 775-825页

作者： Li, Jialun Naud, Frederic Pan, Wenyu Univ Bordeaux Inst Math Talence France Univ Zurich Inst Math Zurich Switzerland Sorbonne Univ Inst Math Jussieu Paris Rive Gauche Paris France Penn State Univ State Coll PA USA Univ Chicago Dept Math Chicago IL 60637 USA

Let Gamma be a Zariski-dense Kleinian Schottky subgroup of PSL2(C). Let Lambda(Gamma) subset of C be its limit set, endowed with a Patterson-Sullivan measure mu supported on Lambda(Gamma). We show that the Fourier transform b (mu) over cap(xi) enjoys polynomial decay as vertical bar xi vertical bar goes to infinity. As a corollary, all limit sets of Zariski-dense Kleinian groups have positive Fourier dimension. This is a PSL2(C) version of the PSL2(R) result of Bourgain and Dyatlov, and uses the decay of exponential sums based on Bourgain-Gamburd's sum-product estimate on C. These bounds on exponential sums require a delicate nonconcentration hypothesis which is proved using some representation theory and regularity estimates for stationary measures of certain random walks on linear groups.

关键词： 42B10 22E40 37C85 additive combinatorics Fourier transform Hausdorff and Fourier dimension transfer operators limit sets of Kleininan groups Patterson–Sullivan measures random walks on linear groups Schottky groups

来源：评论

学校读者我要写书评

暂无评论

On a sumset problem of dilates

引用

INDIAN JOURNAL OF PURE & APPLIED MATHEMATICS 2021年第4期52卷 1180-1185页

作者： Chahal, Sandeep Singh Pandey, Ram Krishna Akal Univ Dept Math Talwandi Sabo 151302 India Indian Inst Technol Roorkee Dept Math Roorkee 247667 Uttar Pradesh India

Let A be a nonempty finite set of integers. For a real number m, the set m . A = {ma : a is an element of A} denotes the set of m-dilates of A. In 2008, Bukh initiated an interesting problem of finding a lower bound for the sumset of dilated sets, i.e., a lower bound for vertical bar m(1) . A + m(2) . A + . . . +m(n) . A vertical bar, where m(1), m(2),..., m(n) are integers. In this paper, we consider the case vertical bar 3 . A + k . A vertical bar, where k is a prime number >= 5. Under some assumptions on A, first we give a general lower bound on the cardinality of 3 . A + k . A then, for large sets A, under the same assumptions, we improve this general lower bound. The results also hold true for the general sumset q . A + k . A, where q is any odd prime

关键词： additive combinatorics Sumsets of dilates

来源：评论

学校读者我要写书评

暂无评论

Inverse problems for minimal complements and maximal supplements

引用

JOURNAL OF NUMBER THEORY 2021年 223卷 307-324页

作者： Alon, Noga Kravitz, Noah Larson, Matt Princeton Univ Dept Math Princeton NJ 08544 USA Tel Aviv Univ Sch Math IL-69978 Tel Aviv Israel Tel Aviv Univ Sch Comp Sci IL-69978 Tel Aviv Israel Zoom Univ Yale Grace Hopper Coll New Haven CT 06511 USA Dept Math 450 Jane Stanford Way Stanford CA 94305 USA

Given a subset W of an abelian group G, a subset C is called an additive complement for W if W + C = G;if, moreover, no proper subset of C has this property, then we say that C is a minimal complement for W. It is natural to ask which subsets C can arise as minimal complements for some W. We show that in a finite abelian group G, every non-empty subset C of size vertical bar C vertical bar <= 2(2)/(3)vertical bar G vertical bar(1/3)/((3e log vertical bar G vertical bar)(2/3) is a minimal complement for some W. As a corollary, we deduce that every finite non-empty subset of an infinite abelian group is a minimal complement. We also derive several analogous results for "dual" problems about maximal supplements. (C) 2020 Elsevier Inc. All rights reserved.

关键词： Minimal complement additive combinatorics Probabilistic combinatorics

来源：评论

学校读者我要写书评

暂无评论

Polynomial Methods: Recent Advancements in combinatorics

Polynomial Methods: Recent Advancements in Combinatorics

引用

作者： Fleming, Thomas The University of Memphis

学位级别：M.S., Master of Science/Master of Surgery

In this Master’s Thesis, we showcase the use of an array of results collectively known as the polynomial method. First, we lay groundwork, giving some basic definitions, notation, and prerequisites. Then, we introduce three related theorems, the Combinatorial Nullstellensatz, Generalized Combinatorial Nullstellensatz, and Punctured Combinatorial Nullstellensatz, each concerning properties of multivariate polynomials with specific constraints on their degree. With these results proven, we then showcase some simple combinatorial and graph theoretic results which have very simple proofs making use of the Nullstellensatz theorems. In chapter 4 we give the proof of the q-Dyson theorem as published by Zeilberger and Bressoud. Finally, in chapters 5, 6 and 7 we introduce zero-sum theory, prove the Davenport constant of finite abelian p-groups without making use of group rings, and then state and prove the Chevalley-Warning theorem and use it to prove the Kemnitz conjecture on zero-sums in cyclic groups.

关键词： additive combinatorics additive number theory Combinatorial number theory combinatorics Number theory Zero-sum theory

来源：评论

学校读者我要写书评

暂无评论

Sets in Zk with doubling 2k+δ are near convex progressions

引用

ADVANCES IN MATHEMATICS 2023年第1期413卷

作者： Van Hintum, Peter Spink, Hunter Tiba, Marius

For delta > 0 sufficiently small and A C Zk with |A + A| < (2k + delta)|A|, we show either A is covered by mk(delta) parallel hyperplanes, or satisfies |-co(A) \ A| < ck delta|A|, where co(A) is the smallest convex progression (convex set intersected with an affine sub-lattice) containing A. This generalizes the Freiman-Bilu 2k theorem, Freiman's 3|A| - 4 theorem, and recent sharp stability results of the present authors for sumsets in Rk conjectured by Figalli and Jerison.(c) 2022 Elsevier Inc. All rights reserved.

关键词： additive combinatorics Convex progressions Small doubling

来源：评论

学校读者我要写书评

暂无评论

Compact sets with large projections and nowhere dense sumset

引用

NONLINEARITY 2023年第10期36卷 5190-5215页

作者： Balka, Richard Elekes, Marton Kiss, Viktor Nagy, Donat Poor, Mark Alfred Renyi Inst Math Realtanoda u 13-15 H-1053 Budapest Hungary Eotvos Lorand Univ Inst Math Pazmany Peter s 1-c H-1117 Budapest Hungary Hebrew Univ Jerusalem Einstein Inst Math Edmond J Safra Campus IL-9190401 Jerusalem Israel

We answer a question of Banakh, Jablonska and Jablonski by showing that for d >= 2 there exists a compact set K subset of R-d such that the projection of K onto each hyperplane is of non-empty interior, but K + K is nowhere dense. The proof relies on a random construction. A natural approach in the proofs is to construct such a K in the unit cube with full projections, that is, such that the projections of K agree with that of the unit cube. We investigate the generalization of these problems for projections onto various dimensional subspaces as well as for l-fold sumsets. We obtain numerous positive and negative results, but also leave open many interesting cases. We also show that in most cases if we have a specific example of such a compact set then actually the generic (in the sense of Baire category) compact set in a suitably chosen space is also an example. Finally, utilizing a computer-aided construction, we show that the compact set in the plane with full projections and nowhere dense sumset can be self-similar.

关键词： projection sumset additive combinatorics random construction generic typical self-similar

来源：评论

学校读者我要写书评

暂无评论

建议与咨询留下您的常用邮箱和电话号码，以便我们向您反馈解决方案和替代方法

时间限定

文献类型

馆藏选择

核心期刊

语言

文献类型

帮助

文字说明：

检索规则说明：

检索范例：

分类表

所选分类

限定检索结果

文献类型

馆藏范围

日期分布

学科分类号

主题

机构

作者

语言

请选择保存的检索档案：

请选择收藏分类：

通借通还

建议与咨询 留下您的常用邮箱和电话号码，以便我们向您反馈解决方案和替代方法

时间限定

文献类型

馆藏选择

核心期刊

语言

文献类型

帮助

文字说明：

检索规则说明：

检索范例：

分类表

所选分类

限定检索结果

文献类型

馆藏范围

日期分布

学科分类号

主题

机构

作者

语言

请选择保存的检索档案： 新增检索档案 确定 取消

请选择收藏分类： 新增自定义分类 确定 取消

通借通还

建议与咨询留下您的常用邮箱和电话号码，以便我们向您反馈解决方案和替代方法

请选择保存的检索档案：

请选择收藏分类：