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SETS WITH SEVERAL CENTERS OF SYMMETRY

INTERNATIONAL JOURNAL OF NUMBER THEORY 2011年第5期7卷 1115-1135页

作者： Freiman, Gregory A. Stanchescu, Yonutz V. Tel Aviv Univ Sch Math Sci IL-69978 Tel Aviv Israel Open Univ Israel IL-43107 Raanana Israel Afeka Acad Coll IL-69107 Tel Aviv Israel

Let A be a finite subset of the group Z(2). Let C = {c(0), c(1),..., c(s-1)} be a finite set of s distinct points in the plane. For every 0 <= i <= s - 1, we define D-i = {a - a' : a is an element of A, a' is an element of A, a + a' = 2c(i)} and R-s(A) = |D-0 boolean OR D-1 boolean OR ... boolean OR Ds-1|. In [1, 2], we found the maximal value of R-s(A) in cases s = 1, s = 2 and s = 3 and studied the structure of A assuming that R-3(A) is equal or close to its maximal value. In this paper, we examine the case of s = 4 centers of symmetry and we find the maximal value of R-4(A). Moreover, in cases when the maximal value is attained, we will describe the structure of extremal sets.

关键词： Inverse additive number theory Kakeya problem symmetry additive combinatorics

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additive BASES IN ABELIAN GROUPS

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INTERNATIONAL JOURNAL OF NUMBER THEORY 2010年第4期6卷 799-809页

作者： Lev, Vsevolod F. Muzychuk, Mikhail E. Pinchasi, Rom Univ Haifa Dept Math IL-36006 Tivon Israel Netanya Acad Coll Dept Math & Computat Sci IL-42365 Netanya Israel Technion Israel Inst Technol Dept Math IL-32000 Haifa Israel

Let G be a finite, non-trivial Abelian group of exponent m, and suppose that B1,..., B(k) are generating subsets of G. We prove that if k > 2m ln log2 vertical bar G vertical bar, then the multiset union B(1) boolean OR ... B(k) forms an additive basis of G;that is, for every g is an element of G, there exist A(1) subset of B(1),..., A(k) subset of B(k) such that g = Sigma(k)(i=1)Sigma(a subset of Ai) a. This generalizes a result of Alon, Linial and Meshulam on the additive bases conjecture. As another step towards proving the conjecture, in the case where B1,..., Bk are finite subsets of a vector space, we obtain lower-bound estimates for the number of distinct values, attained by the sums of the form Sigma(k)(i=1)Sigma(a is an element of Ai) a, where A(i) vary over all subsets of B(i) for each i = 1,..., k. Finally, we establish a surprising relation between the additive bases conjecture and the problem of covering the vertices of a unit cube by translates of a lattice, and present a reformulation of (the strong form of) the conjecture in terms of coverings.

关键词： additive bases additive combinatorics

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Zero-sum subsequences of length kq over finite abelian p-groups

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DISCRETE MATHEMATICS 2016年第1期339卷 399-407页

作者： He, Xiaoyu Harvard Univ Cambridge MA 02138 USA

For a finite abelian group G and a positive integer k, let s(k)(G) denote the smallest integer l is an element of N such that any sequence S of elements of G of length vertical bar S vertical bar >= l has a zerosum subsequence with length k. The celebrated Erdos-Ginzburg-Ziv theorem determines s(n)(C-n) = 2n - 1 for cyclic groups C-n, while Reiher showed in 2007 that s(C-n(2)) = 4n - 3. In this paper we prove for a p-group G with exponent exp(G) = q the upper bound s(kq)(G) <= (k + 2d - 2)q + 3D(G) - 3 whenever k >= d, where d = [D(G/q)] and p is a prime satisfying p >= 2d + 3[D(G/2q)] -3, where D(G) is the Davenport constant of the finite abelian group G. This is the correct order of growth in both k and d. Subject to the same assumptions, we show exact equality s(kq)(G) = kq + D(G) - 1 if k >= p + d and p >= 4d - 2, resolving a case of the conjecture of Gao, Han, Peng, and Sun that S-k (exp(G)) (G) = k exp(G) + D(G) -1 whenever k exp(G) >= D(G). We also obtain a general bound s(kn)(C-n(d)) <= 9kn for n with large prime factors and k sufficiently large. Our methods extend the algebraic method of Kubertin, who proved that s(kq)(C-q(d)) <= (k + Cd-2)q - d if k >= d and q is a prime power. (C) 2015 Elsevier B.V. All rights reserved.

关键词： additive combinatorics Zero-sum subsequences

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New Results for the Growth of Sets of Real Numbers

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DISCRETE & COMPUTATIONAL GEOMETRY 2015年第4期54卷 759-770页

作者： Jones, Timothy G. F. Univ Bristol Sch Math Bristol BS8 1TW Avon England

We use the theory of cross ratios to construct a real-valued function f of only three variables with the property that for any finite set A of reals, the set has cardinality at least , for an absolute constant C. Previously-known functions with this property have all been of four variables. We also improve on the state of the art for functions of four variables by constructing a function g for which g(A) has cardinality at least;the previously best-achieved bound was . Finally, we give an example of a five-variable function h for which h(A) has cardinality at least . Proving these results depends only on the Szemer,di-Trotter incidence theorem and an analoguous result for planes due to Edelsbrunner, Guibas and Sharir, each applied in the Erlangen-type framework of Elekes and Sharir. In particular the proofs do not employ the Guth-Katz polynomial partitioning technique or the theory of ruled surfaces. Although the growth exponents for f, g and h are stronger than those for previously-considered functions, it is not clear that they are necessarily sharp. So we pose a question as to whether the bounds on the cardinalities of f(A), g(A) and h(A) can be further strengthened.

关键词： additive combinatorics Expander functions Cross ratios

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A polynomial bound for the arithmetic k-cycle removal lemma in vector spaces

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JOURNAL OF COMBINATORIAL THEORY SERIES A 2018年 160卷 186-201页

作者： Fox, Jacob Lovasz, Laszlo Miklos Sauermann, Lisa Stanford Univ Dept Math Stanford CA 94305 USA UCLA Dept Math Los Angeles CA 90095 USA

For each k >= 3, Green proved an arithmetic k-cycle removal lemma for any abelian group G. The best known bounds relating the parameters in the lemma for general G are of tower-type. For k > 3, even in the case G = F2(n)no better bounds were known prior to this paper. This special case has received considerable attention due to its close connection to property testing of boolean functions. For every k >= 3, we prove a polynomial bound relating the parameters for G = F-p(n)) , where p is any fixed prime. This extends the result for k = 3 by the first two authors. Due to substantial issues with generalizing the proof of the k = 3 case, a new strategy is developed in order to prove the result for k > 3. (C) 2018 Elsevier Inc. All rights reserved.

关键词： additive combinatorics Arithmetic property testing Arithmetic removal lemma Extremal combinatorics

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Near-misses in Wilf's conjecture

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SEMIGROUP FORUM 2019年第2期98卷 285-298页

作者： Eliahou, Shalom Fromentin, Jean Univ Littoral Cote dOpale EA 2597 LMPA Lab Math Pures & Appl Joseph Liouville F-62228 Calais France CNRS FR 2956 Lille Lille France

Let SN be a numerical semigroup with multiplicity m, conductor c and minimal generating set P. Let L=S[0,c-1] and W(S)=|P||L|-c. In 1978, Herbert Wilf asked whether W(S)0 always holds, a question known as Wilf's conjecture and open since then. A related number W0(S), satisfying W0(S)W(S), has recently been introduced. We say that S is a near-miss in Wilf's conjecture if W0(S)<0. Near-misses are very rare. Here we construct infinite families of them, with c=4m and W0(S) arbitrarily small, and we show that the members of these families still satisfy Wilf's conjecture.

关键词： Numerical semigroup Conductor Apery element Sidon set additive combinatorics

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Note on the Theorem of Balog, Szemerédi, and Gowers

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COMBINATORICA 2024年第3期44卷 691-698页

作者： Reiher, Christian Schoen, Tomasz Univ Hamburg Fachbereich Math Hamburg Germany Adam Mickiewicz Univ Dept Discrete Math Poznan Poland

We prove that every additive set A with energy E ( A ) >= | A | 3 / K \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E(A)\ge |A|<^>3/K$$\end{document} has a subset A ' subset of A \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A'\subseteq A$$\end{document} of size | A ' | >= ( 1 - epsilon ) K - 1 / 2 | A | \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|A'|\ge (1-\varepsilon )K<^>{-1/2}|A|$$\end{document} such that | A ' - A ' | <= O epsilon ( K 4 | A ' | ) \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|A'-A'|\le O_\varepsilon (K<^>{4}|A'|)$$\end{document} . This is, essentially, the largest structured set one can get in the Balog-Szemer & eacute;di-Gowers theorem.

关键词： additive combinatorics additive sets of large energy Balog-Szemer & eacute di-Gowers theorem

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On the concentration of points of polynomial maps and applications

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MATHEMATISCHE ZEITSCHRIFT 2012年第3-4期272卷 825-837页

作者： Cilleruelo, Javier Garaev, Moubariz Z. Ostafe, Alina Shparlinski, Igor E. Macquarie Univ Dept Comp Sydney NSW 2109 Australia Univ Nacl Autonoma Mexico Ctr Ciencias Matemat Morelia 58089 Michoacan Mexico CSIC UAM UC3M UCM Inst Ciencias Matemat Madrid 28049 Spain Univ Autonoma Madrid Dept Matemat E-28049 Madrid Spain

For a polynomial f is an element of F-p[X] we obtain upper bounds on the number of points (x, f (x)) modulo a prime p which belong to an arbitrary square with the side length H. Our results in particular are based on the Vinogradov mean value theorem. Using these estimates we obtain results on the expansion of orbits in dynamical systems generated by nonlinear polynomials and we obtain an asymptotic formula for the number of visible points on the curve f(x) equivalent to y (mod p), where f is an element of F-p[X] is a polynomial of degree d >= 2. We also use some recent results and techniques from arithmetic combinatorics to study the values (x, f (x)) in more general sets.

关键词： Polynomial congruences Vinogradov mean value theorem additive combinatorics Orbits Visible points

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Small doubling in prime-order groups: From 2.4 to 2.6

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JOURNAL OF NUMBER THEORY 2020年 217卷 278-291页

作者： Lev, Vsevolod F. Shkredov, Ilya D. Univ Haifa Dept Math IL-36006 Tivon Israel Steklov Math Inst Ul Gubkina 8 Moscow 119991 Russia

Improving upon the results of Freiman and Candela-Serra-Spiegel, we show that for a non-empty subset A subset of F-p with p prime and vertical bar A vertical bar 100, then A is contained in an arithmetic progression ... 详细信息

Improving upon the results of Freiman and Candela-Serra-Spiegel, we show that for a non-empty subset A subset of F-p with p prime and vertical bar A vertical bar < 0.0045p, (i) if vertical bar A + A vertical bar < 2.59 vertical bar A vertical bar - 3 and vertical bar A vertical bar > 100, then A is contained in an arithmetic progression of size vertical bar A + A vertical bar - vertical bar A vertical bar + 1, and (ii) if vertical bar A - A vertical bar < 2.6 vertical bar A vertical bar-3, then A is contained in an arithmetic progression of size vertical bar A - A vertical bar - vertical bar A vertical bar + 1. The improvement comes from using the properties of higher energies. (C) 2020 Elsevier Inc. All rights reserved.

关键词： Sumset additive combinatorics Small doubling

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Sumsets of semiconvex sets

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CANADIAN MATHEMATICAL BULLETIN-BULLETIN CANADIEN DE MATHEMATIQUES 2022年第1期65卷 84-94页

作者： Ruzsa, Imre Solymosi, Jozsef Alfred Renyi Inst Math Budapest Hungary Univ British Columbia Dept Math Vancouver BC Canada

We investigate additive properties of sets A, where A = {a(1), a(2), ..., a(k)} is a monotone increasing set of real numbers, and the differences of consecutive elements are all distinct. It is known that vertical bar A + B vertical bar >= c vertical bar A vertical bar vertical bar B vertical bar(1/2) for any finite set of numbers B. The bound is tight up to the constant multiplier. We give a new proof to this result using bounds on crossing numbers of geometric graphs. We construct examples showing the limits of possible improvements. In particular, we show that there are arbitrarily large sets with different consecutive differences and sub-quadratic sumset size.

关键词： sumset inequalities convex sets of numbers additive combinatorics

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