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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

### Centre de Recherches Mathématiques CRM Proceedings and Lecture Notes Volume??, 2007 Ergodic Methods in Additive Combinatorics

"... Abstract. Shortly after Szemerédi’s proof that a set of positive upper density contains arbitrarily long arithmetic progressions, Furstenberg gave a new proof of this theorem using ergodic theory. This gave rise to the field of combinatorial ergodic theory, in which problems motivated by additive co ..."

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Abstract. Shortly after Szemerédi’s proof that a set of positive upper density contains arbitrarily long arithmetic progressions, Furstenberg gave a new proof of this theorem using ergodic theory. This gave rise to the field of combinatorial ergodic theory, in which problems motivated by additive combinatorics are addressed kwith ergodic theory. Combinatorial ergodic theory has since produced combinatorial results, some of which have yet to be obtained by other means, and has also given a deeper understanding of the structure of measure preserving systems. We outline the ergodic theory background needed to understand these results, with an emphasis on recent developments in ergodic theory and the relation to recent developments in additive combinatorics. These notes are based on four lectures given during the School on Additive Combinatorics at the Centre de recherches mathématiques, Montreal in April, 2006. The talks were aimed at an audience without background in ergodic theory. No attempt is made to include complete proofs of all statements and often the reader is referred to the original sources. Many of the proofs included are classic, included as an indication of which ingredients play a role in the developments of the past ten years. 1. Combinatorics to ergodic theory 1.1. Szemerédi’s theorem. Answering a long standing conjecture of Erdős and Turán [11], Szemerédi [54] showed that a set E ⊂ Z with positive upper density 1 contains arbitrarily long arithmetic progressions. Soon thereafter, Furstenberg [16] gave a new proof of Szemerédi’s Theorem using ergodic theory, and this has lead to the rich field of combinatorial ergodic theory. Before describing some of the results in this subject, we motivate the use of ergodic theory for studying combinatorial problems. We start with the finite formulation of Szemerédi’s theorem: Theorem 1.1 (Szemerédi [54]). Given δ> 0 and k ∈ N, there is a function N(δ, k) such that if N> N(δ, k) and E ⊂ {1,..., N} is a subset with |E | ≥ δN, then E contains an arithmetic progression of length k.

### FROM HARMONIC ANALYSIS TO ARITHMETIC COMBINATORICS

"... Abstract. We will describe a certain line of research connecting classical harmonic analysis to PDE regularity estimates, an old question in Euclidean geometry, a variety of deep combinatorial problems, recent advances in analytic number theory, and more. Traditionally, restriction theory is a part ..."

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Abstract. We will describe a certain line of research connecting classical harmonic analysis to PDE regularity estimates, an old question in Euclidean geometry, a variety of deep combinatorial problems, recent advances in analytic number theory, and more. Traditionally, restriction theory is a part of classical Fourier analysis that investigates the relationship between geometric and Fourier-analytic properties of singular measures. It became clear over the years that the theory would have to involve sophisticated geometric and combinatorial input. Two particularly important turning points were Fefferman’s work in the 1970s invoking the ”Kakeya problem ” in this context, and Bourgain’s application of Gowers’s additive number theory techniques to the Kakeya problem almost 30 years later. All this led harmonic analysts to explore areas previously foreign to them, such as combinatorial geometry, graph theory, and additive number theory. Although the Kakeya and restriction problems remain stubbornly open, the