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The primes contain arbitrarily long arithmetic progressions
 Ann. of Math
"... Abstract. We prove that there are arbitrarily long arithmetic progressions of primes. ..."
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Abstract. We prove that there are arbitrarily long arithmetic progressions of primes.
Problems and Results on Combinatorial Number Theory
 J. N. SRIVASTAVA ET AL., EDS., A SURVEY OF COMBINATORIAL THEORY OC NORTHHOLLAND PUBLISHING COMPANY, 1973
, 1973
"... I will discuss in this paper number theoretic problems which are of combinatorial nature. I certainly do not claim to cover the field completely and the paper will be biased heavily towards problems considered by me and my collaborators. Combinatorial methods have often been used successfully in num ..."
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Cited by 16 (1 self)
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I will discuss in this paper number theoretic problems which are of combinatorial nature. I certainly do not claim to cover the field completely and the paper will be biased heavily towards problems considered by me and my collaborators. Combinatorial methods have often been used successfully in number theory (e.g. sieve methods), but here we will try to restrict ourselves to problems which themselves have a combinatorial flavor. I have written several papers in recent years on such problems and in order to avoid making this paper too long, wherever possible, will discuss either problems not mentioned in the earlier papers or problems where some progress has been made since these papers were written. Before starting the discussion of our problems I give a few of the principal papers where similar problems were discussed and where further literature can be found.
Shortened array codes of large girth,” in
 IEEE Transactions on Information Theory
, 2006
"... Abstract — One approach to designing structured lowdensity paritycheck (LDPC) codes with large girth is to shorten codes with small girth in such a manner that the deleted columns of the paritycheck matrix contain all the variables involved in short cycles. This approach is especially effective i ..."
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Abstract — One approach to designing structured lowdensity paritycheck (LDPC) codes with large girth is to shorten codes with small girth in such a manner that the deleted columns of the paritycheck matrix contain all the variables involved in short cycles. This approach is especially effective if the paritycheck matrix of a code is a matrix composed of blocks of circulant permutation matrices, as is the case for the class of codes known as array codes. We show how to shorten array codes by deleting certain columns of their paritycheck matrices so as to increase their girth. The shortening approach is based on the observation that for array codes, and in fact for a slightly more general class of LDPC codes, the cycles in the corresponding Tanner graph are governed by certain homogeneous linear equations with integer coefficients. Consequently, we can selectively eliminate cycles from an array code by only retaining those columns from the paritycheck matrix of the original code that are indexed by integer sequences that do not contain solutions to the equations governing those cycles. We provide Ramseytheoretic estimates for the maximum number of columns that can be retained from the original paritycheck matrix with the property that the sequence of their indices avoid solutions to various types of cyclegoverning equations. This translates to estimates of the rate penalty incurred in shortening a code to eliminate cycles. Simulation results show that for the codes considered, shortening them to increase the girth can lead to significant gains in signaltonoise ratio in the case of communication over an additive white Gaussian noise channel. Index Terms — Array codes, LDPC codes, shortening, cyclegoverning equations
Linear equation, arithmetic progressions and hypergraph property testing
 Proc. of the 16 th Annual ACMSIAM SODA, ACM Press
, 2005
"... For a fixed kuniform hypergraph D (kgraph for short, k ≥ 3), we say that a kgraph H) if it contains no copy (resp. induced copy) of D. Our goal in satisfies property PD (resp. P ∗ D this paper is to classify the kgraphs D for which there are propertytesters for testing PD and P ∗ D whose query ..."
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For a fixed kuniform hypergraph D (kgraph for short, k ≥ 3), we say that a kgraph H) if it contains no copy (resp. induced copy) of D. Our goal in satisfies property PD (resp. P ∗ D this paper is to classify the kgraphs D for which there are propertytesters for testing PD and P ∗ D whose query complexity is polynomial in 1/ɛ. For such kgraphs we say that PD (resp. P ∗ D) is easily testable. For P ∗ D, we prove that aside from a single 3graph, P ∗ D is easily testable if and only if D is a single kedge. We further show that for large k, one can use more sophisticated techniques in order to obtain better lower bounds for any large enough kgraph. These results extend and improve previous results about graphs [5] and kgraphs [18]. For PD, we show that for any kpartite kgraph D, PD is easily testable, by giving an efficient onesided errorproperty tester, which improves the one obtained by [18]. We further prove a nearly matching lower bound on the query complexity of such a propertytester. Finally, we give a sufficient condition for inferring that PD is not easily testable. Though our results do not supply a complete characterization of the kgraphs for which PD is easily testable, they are a natural
The multiparty communication complexity of ExactT: Improved bounds and new problems
 In Proc. of 31st MFCS
, 2006
"... Abstract Let x1,..., xk be nbit numbers and T 2 N. Assume that P1,..., Pk are players such that Pi knows all of the numbers except xi. The players want to determine if Pkj=1 xj = T bybroadcasting as few bits as possible. Chandra, Furst, and Lipton obtained an upper bound of O(pn) bits for the k = 3 ..."
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Abstract Let x1,..., xk be nbit numbers and T 2 N. Assume that P1,..., Pk are players such that Pi knows all of the numbers except xi. The players want to determine if Pkj=1 xj = T bybroadcasting as few bits as possible. Chandra, Furst, and Lipton obtained an upper bound of O(pn) bits for the k = 3 case, and a lower bound of!(1) for k> = 3 when T = \Theta (2n). We obtain(1) for general k> = 3 an upper bound of k + O(n1/(k1)), (2) for k = 3, T = \Theta (2n), a lowerbound of \Omega (log log n), (3) a generalization of the protocol to abelian groups, (4) lower boundson the multiparty communication complexity of some regular languages, (5) lower bounds on branching programs, and (6) empirical results for the k = 3 case. 1 Introduction Multiparty communication complexity was first defined by Chandra, Furst, and Lipton [8] and used to obtain lower bounds on branching programs. Since then it has been used to get additional lower bounds and tradeoffs for branching programs [1, 5], lower bounds on problems in data structures [5], timespace tradeoffs for restricted Turing machines [1], and unconditional pseudorandom generators for logspace [1]. Def 1.1 Let f: {{0, 1}n}k! {0, 1}. Assume, for 1 < = i < = k, Pi has all of the inputs except xi. Let d(f) be the total number of bits broadcast in the optimal deterministic protocol for f. This is called the multiparty communication complexity of f. The scenario is called the forehead model.
Behrendtype constructions for sets of linear equations
"... A linear equation on k unknowns is called a (k, h)equation if it is of the form � k i=1 aixi = 0, with ai ∈ {−h,..., h} and � ai = 0. For a (k, h)equation E, let rE(n) denote the size of the largest subset of the first n integers with no solution of E (besides certain trivial solutions). Several s ..."
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A linear equation on k unknowns is called a (k, h)equation if it is of the form � k i=1 aixi = 0, with ai ∈ {−h,..., h} and � ai = 0. For a (k, h)equation E, let rE(n) denote the size of the largest subset of the first n integers with no solution of E (besides certain trivial solutions). Several special cases of this general problem, such as Sidon’s equation and sets without threeterm arithmetic progressions, are some of the most well studied problems in additive number theory. Ruzsa was the first to address the general problem of the influence of certain properties of equations on rE(n). His results suggest, but do not imply, that for every fixed k, all but an O(1/h) fraction of the (k, h)equations E are such that rE(n)> n 1−o(1). In this paper we address the generalized problem of estimating the size of the largest subset of the first n integers with no solution of a set S of (k, h)equations (again, besides certain trivial solutions). We denote this quantity by rS(n). Our main result is that all but an O(1/h) fraction of the sets of (k, h)equations S of size k − ⌊ √ 2k ⌋ + 1, are such that rS(n)> n 1−o(1). We also give several additional results relating properties of sets of equations and rS(n).
Finding Large Sets Without Arithmetic Progressions of Length Three: An Empirical View and Survey II
, 2010
"... There has been much work on the following question: given n, how large can a subset of {1,..., n} be that has no arithmetic progressions of length 3. We call such sets 3free. Most of the work has been asymptotic. In this paper we sketch applications of large 3free sets, review the literature of ho ..."
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There has been much work on the following question: given n, how large can a subset of {1,..., n} be that has no arithmetic progressions of length 3. We call such sets 3free. Most of the work has been asymptotic. In this paper we sketch applications of large 3free sets, review the literature of how to construct large 3free sets, and present empirical studies on how large such sets actually are. The two main questions considered are (1) How large can a 3free set be when n is small, and (2) How do the methods in the literature compare to each other? In particular,
Long arithmetic progressions of primes
 Mathematics Proceedings
"... Abstract. This is an article for a general mathematical audience on the author’s work, joint with Terence Tao, establishing that there are arbitrarily long arithmetic progressions of primes. 1. introduction and history This is a description of recent work of the author and Terence Tao [11] on primes ..."
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Abstract. This is an article for a general mathematical audience on the author’s work, joint with Terence Tao, establishing that there are arbitrarily long arithmetic progressions of primes. 1. introduction and history This is a description of recent work of the author and Terence Tao [11] on primes in arithmetic progression. It is based on seminars given for a general mathematical
ON A TWO–DIMENSIONAL ANALOG OF SZEMER ÉDI’S THEOREM IN ABELIAN GROUPS
, 705
"... Let G be a finite Abelian group and A ⊆ G × G be a set of cardinality at least G  2 /(log log G) c, where c> 0 is an absolute constant. We prove that A contains a triple {(k, m), (k + d, m), (k, m+ d)}, where d ̸ = 0. This theorem is a twodimensional generalization of Szemerédi’s theorem on ari ..."
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Let G be a finite Abelian group and A ⊆ G × G be a set of cardinality at least G  2 /(log log G) c, where c> 0 is an absolute constant. We prove that A contains a triple {(k, m), (k + d, m), (k, m+ d)}, where d ̸ = 0. This theorem is a twodimensional generalization of Szemerédi’s theorem on arithmetic progressions. 1. Introduction. Szemerédi’s theorem [29] on arithmetic progressions states that an arbitrary set A ⊆ Z of positive density contains arithmetic progression of any length. This remarkable theorem has played a significant role in the development of two fields in mathematics: additive combinatorics (see e.g. [31]) and combinatorial ergodic