New bounds on cap sets

Bateman, Michael; Katz, Nets Hawk

doi:10.1090/s0894-0347-2011-00725-x

Cited by 68 publications

(184 citation statements)

References 7 publications

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“…Meshulam [M95], following the general lines of Roth's argument, has shown that if G is an abelian group of odd order, then r 3 (G) ≤ 2|G|/ rk(G) (where we use the standard notation rk(G) for the rank of G); in particular, r 3 (Z n m ) ≤ 2m n /n. Despite many efforts, no further progress was made for over 15 years, till Bateman and Katz in their ground-breaking paper [BK12] proved that r 3 (Z n 3 ) = O(3 n /n 1+ε ) with an absolute constant ε > 0.…”

Section: Background and Motivationmentioning

confidence: 99%

Progression-free sets in $\mathbb{Z}_4^n$ are exponentially small

2017

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Abstract. We show that for integer n ≥ 1, any subset A ⊆ Z n 4 free of three-term arithmetic progressions has size |A| ≤ 4 γn , with an absolute constant γ ≈ 0.926. Background and MotivationIn his influential papers [R52, R53], Roth has shown that if a set A ⊆ {1, 2, . . . , N } does not contain three elements in an arithmetic progression, then |A| = o(N ) and indeed, |A| = O(N/ log log N ) as N grows. Since then, estimating the largest possible size of such a set has become one of the central problems in additive combinatorics. 4 / log N ). It is easily seen that Roth's problem is essentially equivalent to estimating the largest possible size of a subset of the cyclic group Z N , free of three-term arithmetic progressions. This makes it natural to investigate other finite abelian groups.We say that a subset A of an (additively written) abelian group G is progression-free if there do not exist pairwise distinct a, b, c ∈ A with a + b = 2c, and we denote by r 3 (G) the largest size of a progression-free subset A ⊆ G. For abelian groups G of odd order, Brown and Buhler [BB82] and independently Frankl, Graham, and Rödl [FGR87] proved that r 3 (G) = o(|G|) as |G| grows. Meshulam [M95], following the general lines of Roth's argument, has shown that if G is an abelian group of odd order, then r 3 (G) ≤ 2|G|/ rk(G) (where we use the standard notation rk(G) for the rank of G); in particular, r 3 (Z n m ) ≤ 2m n /n. Despite many efforts, no further progress was made for over 15 years, till Bateman and Katz in their ground-breaking paper [BK12] proved that r 3 (Z n 3 ) = O(3 n /n 1+ε ) with an absolute constant ε > 0.Abelian groups of even order were first considered in [L04] where, as a further elaboration on the Roth-Meshulam proof, it is shown that r 3 (G) < 2|G|/ rk(2G) for any finite abelian group G; here 2G = {2g : g ∈ G}. For the homocyclic groups of exponent 4 this †

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Section: Background and Motivationmentioning

confidence: 99%

Progression-free sets in $\mathbb{Z}_4^n$ are exponentially small

2017

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“…Meshulam [12], following the general lines of Roth's argument, has shown that if G is an abelian group of odd order, then r 3 (G)≤2|G|/rk(G) (where we use the standard notation rk(G) for the rank of G). Despite many efforts, no further progress was made for over 15 years, till Bateman and Katz [163] in their ground-breaking paper proved that r 3 (Z 3 n ) = O(3 n / n 1+ε ) with an absolute constant ε>0. Abelian groups of even order were first considered by Lev in 2004, who, as a further elaboration on the Roth-Meshulam proof, has shown that r 3 (G)<2|G|/rk(2G) for any finite abelian group G; here 2G = {2g: g is in G}.…”

Section: Combinatorial Resultsmentioning

confidence: 99%

BME VIK Annual Research Report on Electrical Engineering and Computer Science 2016

Charaf

Harsányi

Poppe

et al. 2017

Period. Polytech. Elec. Eng. Comp. Sci.

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Preface Since being established in 1949, the Faculty of Electrical Engineering and Informatics (VIK) BME has played a flagship role in the development of electronics, IT and computer science in Hungary.We László Jakab (dean, BME VIK) János Levendovszky (vice-dean in charge of scientific affairs, BME VIK) are proud of combining engineering applications with sound scientific results, which is the assurance of high-level industrial collaboration leading to novel results and innovation. The various collaborations with our industrial partners has made it clear that the industry expects methods and results which make their industrial processes more effective and increase productivity and quality. Thus, participation in these collaborations give a competitive edge and ensure the continuous development of VIK. These factors have positioned our Department of Automation and Applied InformaticsDepartment of Automation and Applied Informatics (AUT) is one of the largest and dynamically developing departments at BME with diverse scope competences such as control theory, embedded systems, software modelling, applied software development, Internet of Things, power electronics, mechatronics, and many others. The department's main activities are education, research and development. Training versatile electrical and software engineers with solid practical knowledge is our top priority. Our profile also includes developing high quality software and hardware solutions for industry partners. All of our activities are backed by strong research background.In 2016, the department organized and hosted a number of domestic and international events. In cooperation with the industry, AUT has organized the IoT4U conference. It was dedicated to Internet of Things, cloud and data management as well as the integration of IoT into various formats. RobonAUT, the yearly domestic amateur robotics contest was organized by the department in February 2016. The yearly contest is open to teams of students, organized either in student projects within the department or independent research teams.The next sections summarize the key research results of the department in year 2016. Software Development Methods and Mobile PlatformsThe diversity of mobile platforms necessitates the development of the same mobile application for all major mobile platforms, what requires considerable development effort. Mobile application developers are multiplatform developers, but they still prioritize the platforms, therefore, not all platforms are equally important for them. Appropriate methods, processes and tools are required to support the development in order to achieve better productivity.The research on educational mobile applications that adapt to the mental state of the user required a method for estimating the difficulty of tasks found in the serious games. There are generated or random assignments where the effort required from the user is not known or cannot be estimated in advance.61 (2) IoT and Smart CityThe Internet of Things (IoT) is transforming the surroun...

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“…2 Note, however, that the problem of counting a 3-term arithmetic progressions becomes trivial in characteristic 2, as it amounts to computing the number of triples (x, y, z) ∈ A 3 satisfying x + y = 2z ≡ 0 mod 2. 3 Note that while any 3-term progression in F n 3 forms a complete line, the proof makes no use of this special geometric feature.…”

Section: The Classicsmentioning

confidence: 99%

Finite field models in arithmetic combinatorics – ten years on

Wolf

2015

Finite Fields and Their Applications

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General rightsThis document is made available in accordance with publisher policies. Please cite only the published version using the reference above. Full terms of use are available: http://www.bristol.ac.uk/pure/about/ebr-termsAbstract. It has been close to ten years since the publication of Green's influential survey Finite field models in additive combinatorics [?], in which the author championed the use of high-dimensional vector spaces over finite fields as a toy model for tackling additive problems concerning the integers. The path laid out by Green has proven to be a very successful one to follow. In the present article we survey the highlights of the past decade and outline the challenges for the years to come.

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New bounds on cap sets

Cited by 68 publications

References 7 publications

Progression-free sets in $\mathbb{Z}_4^n$ are exponentially small

Progression-free sets in $\mathbb{Z}_4^n$ are exponentially small

BME VIK Annual Research Report on Electrical Engineering and Computer Science 2016

Finite field models in arithmetic combinatorics – ten years on

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