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The dichotomy between structure and randomness, arithmetic progressions, and the primes
"... Abstract. A famous theorem of Szemerédi asserts that all subsets of the integers with positive upper density will contain arbitrarily long arithmetic progressions. There are many different proofs of this deep theorem, but they are all based on a fundamental dichotomy between structure and randomness ..."
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Abstract. A famous theorem of Szemerédi asserts that all subsets of the integers with positive upper density will contain arbitrarily long arithmetic progressions. There are many different proofs of this deep theorem, but they are all based on a fundamental dichotomy between structure and randomness, which in turn leads (roughly speaking) to a decomposition of any object into a structured (lowcomplexity) component and a random (discorrelated) component. Important examples of these types of decompositions include the Furstenberg structure theorem and the Szemerédi regularity lemma. One recent application of this dichotomy is the result of Green and Tao establishing that the prime numbers contain arbitrarily long arithmetic progressions (despite having density zero in the integers). The power of this dichotomy is evidenced by the fact that the GreenTao theorem requires surprisingly little technology from analytic number theory, relying instead almost exclusively on manifestations of this dichotomy such as Szemerédi’s theorem. In this paper we survey various manifestations of this dichotomy in combinatorics, harmonic analysis, ergodic theory, and number theory. As we hope to emphasize here, the underlying themes in these arguments are remarkably similar even though the contexts are radically different. 1.
Ergodic theory and configurations in sets of positive density
 Mathematics of Ramsey theory, 184–198, Algorithms Combin
, 1990
"... We shall present here two examples from "geometric Ramsey theory " which illustrate how ergodic theoretic techniques can be used to prove that subsets of Euclidean space of positive density necessarily contain certain configurations. Specifically we will deal with subsets of the plane, and ..."
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We shall present here two examples from "geometric Ramsey theory " which illustrate how ergodic theoretic techniques can be used to prove that subsets of Euclidean space of positive density necessarily contain certain configurations. Specifically we will deal with subsets of the plane, and our results will be valid
The sovability of norm, bilinear and quadratic equations over finite fields via spectra of graphs
, 2009
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Pinned distance sets, ksimplices, Wolff’s exponent in finite fields and sumproduct estimates
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Distance sets corresponding to convex bodies
, 2003
"... Suppose that K ⊆ R d is a 0symmetric convex body which defines the usual norm ‖x‖ K = sup {t ≥ 0: x / ∈ tK} on R d. Let also A ⊆ R d be a measurable set of positive upper density ρ. We show that if the body K is not a polytope, or if it is a polytope with many faces (depending on ρ), then the dista ..."
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Suppose that K ⊆ R d is a 0symmetric convex body which defines the usual norm ‖x‖ K = sup {t ≥ 0: x / ∈ tK} on R d. Let also A ⊆ R d be a measurable set of positive upper density ρ. We show that if the body K is not a polytope, or if it is a polytope with many faces (depending on ρ), then the distance set DK(A) = {‖x − y‖ K: x, y ∈ A} contains all points t ≥ t0 for some positive number t0. This was proved by Katznelson and Weiss, by Falconer and Marstrand and by Bourgain in the case where K is the Euclidean ball in any dimension. As corollaries we obtain (a) an extension to any dimension of a theorem of Iosevich and Łaba regarding distance sets with respect to convex bodies of welldistributed sets in the plane, and also (b) a new proof of a theorem of Iosevich, Katz and Tao about the nonexistence of Fourier spectra for smooth convex bodies.
On ksimplexes in (2k − 1)dimensional vector spaces over finite fields
, 2009
"... We show that if the cardinality of a subset of the (2k − 1)dimensional vector 1 2k−1− space over a finite field with q elements is ≫ q 2k, then it contains a positive proportional of all ksimplexes up to congruence. ..."
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We show that if the cardinality of a subset of the (2k − 1)dimensional vector 1 2k−1− space over a finite field with q elements is ≫ q 2k, then it contains a positive proportional of all ksimplexes up to congruence.
An optimal version of Sárközy’s theorem
"... Abstract. Using Fourier analytic techniques, we prove that if ε> 0, N ≥ exp exp(Cε−1 log ε−1) and A ⊆ {1,..., N}, then there must exist t ∈ N such that A ∩ (A + t2) N ..."
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Abstract. Using Fourier analytic techniques, we prove that if ε> 0, N ≥ exp exp(Cε−1 log ε−1) and A ⊆ {1,..., N}, then there must exist t ∈ N such that A ∩ (A + t2) N