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55
The ergodic and combinatorial approaches to Szemerédi’s theorem
, 2006
"... A famous theorem of Szemerédi asserts that any set of integers of positive upper density will contain arbitrarily long arithmetic progressions. In its full generality, we know of four types of arguments that can prove this theorem: the original combinatorial (and graph-theoretical) approach of Szem ..."
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Cited by 17 (2 self)
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A famous theorem of Szemerédi asserts that any set of integers of positive upper density will contain arbitrarily long arithmetic progressions. In its full generality, we know of four types of arguments that can prove this theorem: the original combinatorial (and graph-theoretical) approach of Szemerédi, the ergodic theory approach of Furstenberg, the Fourier-analytic approach of Gowers, and the hypergraph approach of Nagle-Rödl-Schacht-Skokan and Gowers. In this lecture series we introduce the first, second and fourth approaches, though we will not delve into the full details of any of them. One of the themes of these lectures is the strong similarity of ideas between these approaches, despite the fact that they initially seem rather different.
DECOMPOSITIONS, APPROXIMATE STRUCTURE, TRANSFERENCE, AND THE HAHN-BANACH THEOREM
, 2008
"... We discuss three major classes of theorems in additive and extremal combinatorics: decomposition theorems, approximate structure theorems, and transference principles. We also show how the finite-dimensional Hahn-Banach theorem can be used to give short and transparent proofs of many results of the ..."
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We discuss three major classes of theorems in additive and extremal combinatorics: decomposition theorems, approximate structure theorems, and transference principles. We also show how the finite-dimensional Hahn-Banach theorem can be used to give short and transparent proofs of many results of these kinds. Amongst the applications of this method is a much shorter proof of one of the major steps in the proof of Green and Tao that the primes contain arbitrarily long arithmetic progressions. In order to explain the role of this step, we include a brief description of the rest of their argument. A similar proof has been discovered independently by Reingold, Trevisan, Tulsiani and Vadhan [RTTV].
The metamathematics of ergodic theory
- THE ANNALS OF PURE AND APPLIED LOGIC
, 2009
"... The metamathematical tradition, tracing back to Hilbert, employs syntactic modeling to study the methods of contemporary mathematics. A central goal has been, in particular, to explore the extent to which infinitary methods can be understood in computational or otherwise explicit terms. Ergodic theo ..."
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Cited by 13 (1 self)
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The metamathematical tradition, tracing back to Hilbert, employs syntactic modeling to study the methods of contemporary mathematics. A central goal has been, in particular, to explore the extent to which infinitary methods can be understood in computational or otherwise explicit terms. Ergodic theory provides rich opportunities for such analysis. Although the field has its origins in seventeenth century dynamics and nineteenth century statistical mechanics, it employs infinitary, nonconstructive, and structural methods that are characteristically modern. At the same time, computational concerns and recent applications to combinatorics and number theory force us to reconsider the constructive character of the theory and its methods. This paper surveys some recent contributions to the metamathematical study of ergodic theory, focusing on the mean and pointwise ergodic theorems and the Furstenberg structure theorem for measure preserving systems. In particular, I characterize the extent to which these theorems are nonconstructive, and explain how proof-theoretic methods can be used to locate their “constructive content.”
Density theorems and extremal hypergraph problems
- Israel J. Math
"... Abstract. We present alternative proofs of density versions of some combinatorial partition theorems originally obtained by E. Szemerédi, H. Furstenberg, and Y. Katznelson. These proofs are based on an extremal hypergraph result which was recently independently obtained by W. T. Gowers and B. Nagle, ..."
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Cited by 9 (3 self)
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Abstract. We present alternative proofs of density versions of some combinatorial partition theorems originally obtained by E. Szemerédi, H. Furstenberg, and Y. Katznelson. These proofs are based on an extremal hypergraph result which was recently independently obtained by W. T. Gowers and B. Nagle, V. Rödl, M. Schacht, J. Skokan by extending Szemerédi’s regularity lemma to hypergraphs.
What is good mathematics
, 2007
"... Abstract. Some personal thoughts and opinions on what “good quality mathematics” is, and whether one should try to define this term rigorously. As a case study, the story of Szemerédi’s theorem is presented. 1. The many aspects of mathematical quality We all agree that mathematicians should strive t ..."
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Cited by 9 (0 self)
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Abstract. Some personal thoughts and opinions on what “good quality mathematics” is, and whether one should try to define this term rigorously. As a case study, the story of Szemerédi’s theorem is presented. 1. The many aspects of mathematical quality We all agree that mathematicians should strive to produce good mathematics. But how does one define “good mathematics”, and should one even dare to try at all? Let us first consider the former question. Almost immediately one realises that there are many different types of mathematics which could be designated “good”. For instance, “good mathematics ” could refer (in no particular order) to (i) Good mathematical problem-solving (e.g. a major breakthrough on an important mathematical problem); (ii) Good mathematical technique (e.g. a masterful use of existing methods, or the development of new tools); (iii) Good mathematical theory (e.g. a conceptual framework or choice of notation which systematically unifies and generalises an existing body of results);
Deducing the density Hales-Jewett theorem from an infinitary removal lemma
- J. Theoret. Probab
"... removal lemma ..."
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Obstructions to uniformity, and arithmetic patterns in the primes
, 2005
"... In this expository article, we describe the recent approach, motivated by ergodic theory, towards detecting arithmetic patterns in the primes, and in particular establishing in [26] that the primes contain arbitrarily long arithmetic progressions. One of the driving philosophies is to identify prec ..."
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Cited by 8 (5 self)
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In this expository article, we describe the recent approach, motivated by ergodic theory, towards detecting arithmetic patterns in the primes, and in particular establishing in [26] that the primes contain arbitrarily long arithmetic progressions. One of the driving philosophies is to identify precisely what the obstructions could be that prevent the primes (or any other set) from behaving “randomly”, and then either show that the obstructions do not actually occur, or else convert the obstructions into usable structural information on the primes.
