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**11 - 16**of**16**### Additive Combinatorics with a view towards Computer Science and Cryptography An

, 2011

"... Recently, additive combinatorics has blossomed into a vibrant area in mathematical sciences. But it seems to be a difficult area to define – perhaps because of a blend of ideas and techniques from several seemingly unrelated contexts which are used there. One might say that additive combinatorics is ..."

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Recently, additive combinatorics has blossomed into a vibrant area in mathematical sciences. But it seems to be a difficult area to define – perhaps because of a blend of ideas and techniques from several seemingly unrelated contexts which are used there. One might say that additive combinatorics is a branch of mathematics concerning the study of additive structures in sets equipped with a group structure – we may have other structure that interacts with this group structure. This newly emerging field has seen tremendous advances over the last few years, and has recently become a focus of attention among both mathematicians and computer scientists. This fascinating area has been enriched by its formidable links to combinatorics, number theory, harmonic analysis, ergodic theory, and some other branches; all deeply cross-fertilize each other, holding great promise for all of them! There is a considerable number of incredible problems, results, and novel applications in this thriving area. In this exposition, we attempt to provide an illuminating overview of some conspicuous breakthroughs in this captivating field, together with a number of seminal applications to sundry parts of mathematics and some other disciplines, with emphasis on computer science and cryptography.

### THREE TOPICS IN ADDITIVE PRIME NUMBER THEORY

, 710

"... Abstract. We discuss, in varying degrees of detail, three contemporary themes in prime number theory. Topic 1: the work of Goldston, Pintz and Yıldırım on short gaps between primes. Topic 2: the work of Mauduit and Rivat, establishing that 50% of the primes have odd digit sum in base 2. Topic 3: wor ..."

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Abstract. We discuss, in varying degrees of detail, three contemporary themes in prime number theory. Topic 1: the work of Goldston, Pintz and Yıldırım on short gaps between primes. Topic 2: the work of Mauduit and Rivat, establishing that 50% of the primes have odd digit sum in base 2. Topic 3: work of Tao and the author on linear equations in primes. Introduction. These notes are to accompany two lectures I am scheduled to give at the Current Developments in Mathematics conference at Harvard in November 2007. The title of those lectures is ‘A good new millennium for primes’, but I have chosen a rather drier title for these notes for two reasons. Firstly, the title of the lectures was unashamedly stolen (albeit with permission) from Andrew Granville’s entertaining

### On the Foundamental Theorem in Arithmetic Progession of Primes

"... Using Jiang function we prove the foundamental theorem in arithmetic progression of primes [1-3]. The primes contain only k < Pg + 1 long arithmetic progressions, but the primes have no k> Pg + 1 long arithmetic progressions.Terence Tao is recipient of 2006 Fields medal.Green and Tao proved that the ..."

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Using Jiang function we prove the foundamental theorem in arithmetic progression of primes [1-3]. The primes contain only k < Pg + 1 long arithmetic progressions, but the primes have no k> Pg + 1 long arithmetic progressions.Terence Tao is recipient of 2006 Fields medal.Green and Tao proved that the primes contain arbitrarily long arithmetic progressions which is absolutely false[4-9].They do not understand the arithmetic progression of primes [4-15].

### unknown title

"... Using the Jiang function we find the best theory of arbitrarily long arithmetic progressions of primes 1 Theorem. The fundamental theorem in arithmetic progression of primes. We define the arithmetic progression of primes [1-3]. P i�1 � P1 ..."

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Using the Jiang function we find the best theory of arbitrarily long arithmetic progressions of primes 1 Theorem. The fundamental theorem in arithmetic progression of primes. We define the arithmetic progression of primes [1-3]. P i�1 � P1

### A SIMPLE REGULARIZATION OF GRAPHS

, 904

"... Abstract. The well-known regularity lemma of E. Szemerédi for graphs (i.e. 2-uniform hypergraphs) claims that for any graph there exists a vertex partition with the property of quasi-randomness. We give a simple construction of such a partition. It is done just by taking a constant-bounded number of ..."

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Abstract. The well-known regularity lemma of E. Szemerédi for graphs (i.e. 2-uniform hypergraphs) claims that for any graph there exists a vertex partition with the property of quasi-randomness. We give a simple construction of such a partition. It is done just by taking a constant-bounded number of random vertex samplings only one time (thus, iteration-free). Since it is independent from the definition of quasi-randomness, it can be generalized very naturally to hypergraph regularization. In this expository note, we show only a graph case of the paper [5] on hypergraphs, but may help the reader to access [5]. 1.