A generalization of Meshulam’s theorem on subsets of finite abelian groups with no 3-term arithmetic progression

Yu-Ru Liu; Craig V. Spencer

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A generalization of Meshulam’s theorem on subsets of finite abelian groups with no 3-term arithmetic progression

机译：Meshulam定理在没有三项算术级数的有限阿贝尔群的子集上的推广

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Let r ₁, …, r _s be non-zero integers satisfying r ₁ + ⋯ + r _s = 0. Let G @ mathbbZ / k₁ mathbbZÅ¼ÅmathbbZ / k_n mathbbZ{simeq mathbb{Z} / k_1 mathbb{Z}oplus cdots oplus mathbb{Z} / k_n mathbb{Z}} be a finite abelian group with k _i|k _i-1(2 ≤ i ≤ n), and suppose that (r _i, k ₁) = 1(1 ≤ i ≤ s). Let D_r(G){D_{mathbf r}(G)} denote the maximal cardinality of a set A Í G{A subseteq G} which contains no non-trivial solution of r ₁ x ₁ + ⋯ + r _s x _s = 0 with x_i Î A (1 £ i £ s){x_i,in,A (1 le i le s)}. We prove that D_r(G) |G|s-2{D_{mathbf r}(G) ll |G|^{s-2}}. We also apply this result to study problems in finite projective spaces.

机译：令r _{1 ，…，r _{s 为非零整数，满足r _{1 +⋯+ r _{s = 0.让G @ mathbbZ / k _{1 mathbbZÅ¼ÅmathbbZ/ k _{n mathbbZ {simeq mathbb {Z} / k_1 mathbb {Z} oplus cdots oplus mathbb {Z} / k_n mathbb {Z}}是一个具有k _{i | k _{i-1 （2≤i≤n）的有限阿贝尔群，并假设（r _{i ，k _{1 ）= 1（1≤i≤s）。令D _{r （G）{D_ {mathbf r}（G）}表示集合AÍG {Asubseteq G}的最大基数，其中不包含r _{的非平凡解。 1 x _{1 +⋯+ r _{s x _{s = 0，其中x _{i ÎA（ 1£i£s）{x_i，in，A（1 le i le s）}。我们证明D _{r （G） | G | / n s-2 {D_ {mathbf r}（G）ll | G | / n ^ {s -2}}。我们还将这个结果用于研究有限射影空间中的问题。}}}}}}}}}}}}}}}}}

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《Designs, Codes and Cryptography》 |2009年第1期|p.83-91|共9页
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Yu-Ru Liu; Craig V. Spencer;
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1. A generalization of Meshulam's theorem on subsets of finite abelian groups with no 3-term arithmetic progression (II) [J] . Liu Y.R., Spencer C.V., Zhao X. European journal of combinatorics . 2011,第2期

机译：Meshulam定理在没有三项算术级数的有限阿贝尔群的子集上的推广（II）
2. A generalization of Meshulam's theorem on subsets of finite abelian groups with no 3-term arithmetic progression [J] . Yu-Ru Liu, Craig V. Spencer Designs, Codes and Crytography . 2009,第1期

机译：Meshulam定理在没有三项算术级数的有限阿贝尔群的子集上的推广
3. On the maximal number of 3-term arithmetic progressions in subsets of Zeta/p Zeta [J] . Green B, Sisask O Bulletin of the London Mathematical Society . 2008,第6期

机译：关于Zeta / p Zeta子集中的3项算术级数的最大数目
4. Generalizations of the Compactness Theorem and Goedel's Completeness Theorem for Nonstandard Finite Structures [C] . Miklos Ajtai International Conference on Theory and Applications of Models of Computation(TAMC 2007); 20070522-25; Shanghai(CN) . 2007

机译：非标准有限结构的紧性定理和Goedel完备性定理的推广
5. Three contributions to topology, algebraic geometry and representation theory: Homological finiteness of abelian covers, algebraic elliptic cohomology theory and monodromy theorems in the elliptic setting. [D] . Yang, Yaping. 2014

机译：对拓扑，代数几何和表示理论的三个贡献：阿贝尔封面的同调有限性，代数椭圆同调理论和椭圆环境中的一峰定理。
6. On the Basis Theorem for Finite Abelian Groups [O] . Jesse Douglas 1953

机译：有限阿贝尔群的基本定理
7. A GENERALIZATION OF MESHULAM’S THEOREM ON SUBSETS OF FINITE ABELIAN GROUPS WITH NO 3-TERM ARITHMETIC PROGRESSION [O] . Yu-ru Liu, Craig, V. Spencer 2015

机译：无三阶算术进展的有限aBEL群的子集中的meshulam定理的推广
8. A NOTE ON VAN DER WAERDEN'S THEOREM ON ARITHMETIC PROGRESSIONS [R] . John R. Rabung 1970

机译：关于算术进展的范德韦登定理的一个注记

A generalization of Meshulam’s theorem on subsets of finite abelian groups with no 3-term arithmetic progression

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