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39
A PROBABILISTIC TECHNIQUE FOR FINDING ALMOSTPERIODS IN ADDITIVE COMBINATORICS
"... We introduce a new probabilistic technique for finding ‘almostperiods’ of convolutions of subsets of finite groups. This allows us to give probabilistic proofs of two classical results in additive combinatorics: Roth’s theorem on threeterm arithmetic progressions and the existence of long arithmet ..."
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Cited by 32 (3 self)
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We introduce a new probabilistic technique for finding ‘almostperiods’ of convolutions of subsets of finite groups. This allows us to give probabilistic proofs of two classical results in additive combinatorics: Roth’s theorem on threeterm arithmetic progressions and the existence of long arithmetic progressions in sumsets A +B in Zp. The bounds we obtain for these results are not the best ones known—these being established using Fourier analysis—but they are of a somewhat comparable quality, which is unusual for a method that is completely combinatorial. Furthermore, we are able to find long arithmetic progressions in sets A + B even when both A and B have density close to 1 / logp, which is much sparser than has previously been possible.
Tight Lower Bounds for the Size of EpsilonNets
"... According to a well known theorem of Haussler and Welzl (1987), any range space of bounded VCdimension admits an εnet of size O () ..."
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Cited by 22 (1 self)
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According to a well known theorem of Haussler and Welzl (1987), any range space of bounded VCdimension admits an εnet of size O ()
The polymath project: lessons from a successful online collaboration in mathematics
 In Proc. CHI ’11. ACM
, 2011
"... Although science is becoming increasingly collaborative, there are remarkably few success stories of online collaborations between professional scientists that actually result in real discoveries. A notable exception is the Polymath Project, a group of mathematicians who collaborate online to solve ..."
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Cited by 11 (0 self)
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Although science is becoming increasingly collaborative, there are remarkably few success stories of online collaborations between professional scientists that actually result in real discoveries. A notable exception is the Polymath Project, a group of mathematicians who collaborate online to solve open mathematics problems. We provide an indepth descriptive history of Polymath, using data analysis and visualization to elucidate the principles that led to its success, and the difficulties that must be addressed before the project can be scaled up. We find that although a small percentage of users created most of the content, almost all users nevertheless contributed some content that was highly influential to the task at hand. We also find that leadership played an important role in the success of the project. Based on our analysis, we present a set of design suggestions for how future collaborative mathematics sites can encourage and foster newcomer participation. Author Keywords largescale collaboration, online collaborative mathematics, online collaborative science, online communities
Optimal testing of multivariate polynomials over small prime fields
, 2011
"... We consider the problem of testing if a given function f: Fn q → Fq is close to a nvariate degree d polynomial over the finite field Fq of q elements. The natural, lowquery, test for this property would be to pick the smallest dimension t = tq,d ≈ d/q such that every function of degree greater tha ..."
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Cited by 10 (4 self)
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We consider the problem of testing if a given function f: Fn q → Fq is close to a nvariate degree d polynomial over the finite field Fq of q elements. The natural, lowquery, test for this property would be to pick the smallest dimension t = tq,d ≈ d/q such that every function of degree greater than d reveals this feature on some tdimensional affine subspace of Fn q and to test that f when restricted to a random tdimensional affine subspace is a polynomial of degree at most d on this subspace. Such a test makes only q t queries, independent of n. Previous works, by Alon et al. [AKK + 05], and Kaufman and Ron [KR06] and Jutla et al. [JPRZ04], showed that this natural test rejected functions that were Ω(1)far from degree dpolynomials with probability at least Ω(q −t) (the results of [KR06] hold for all fields Fq, while the results of [JPRZ04] hold only for fields of prime order). Thus to get a constant probability of detecting functions that were at constant distance from the space of degree d polynomials, the tests made q 2t queries. Kaufman and Ron also noted that when q is prime, then q t queries are necessary. Thus these tests were off by at least a quadratic factor from known lower bounds. It was unclear if the soundness analysis of these tests were tight and this question relates closely to the task
Graph removal lemmas
 SURVEYS IN COMBINATORICS
, 2013
"... The graph removal lemma states that any graph on n vertices with o(nv(H)) copies of a fixed graph H may be made Hfree by removing o(n²) edges. Despite its innocent appearance, this lemma and its extensions have several important consequences in number theory, discrete geometry, graph theory and com ..."
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Cited by 9 (3 self)
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The graph removal lemma states that any graph on n vertices with o(nv(H)) copies of a fixed graph H may be made Hfree by removing o(n²) edges. Despite its innocent appearance, this lemma and its extensions have several important consequences in number theory, discrete geometry, graph theory and computer science. In this survey we discuss these lemmas, focusing in particular on recent improvements to their quantitative aspects.
The structure of collaborative problem solving in a virtual math team. Paper presented at the iConference 2011
 In
, 2006
"... To develop a science of smallgroup interaction in groupware, we need a method for analyzing the structure of computermediated discourse. Conversation analysis offers an analysis of conversational talk in terms of a fine structure of adjacency pairs and offers some suggestions about longer sequence ..."
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Cited by 6 (3 self)
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To develop a science of smallgroup interaction in groupware, we need a method for analyzing the structure of computermediated discourse. Conversation analysis offers an analysis of conversational talk in terms of a fine structure of adjacency pairs and offers some suggestions about longer sequences built on these pairs. This paper presents a case study of students solving a math problem in an online chat environment. It shows that their problemsolving discourse consists of a sequence of exchanges, each built on a base adjacency pair and each contributing a move in the solution process.