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15
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.
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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Cited by 5 (1 self)
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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.
Measurable sets with excluded distances ∗
, 2008
"... For a set of distances D = {d1,..., dk} a set A is called Davoiding if no pair of points of A is at distance di for some i. We show that the density of A is exponentially small in k provided the ratios d1/d2, d2/d3,..., dk−1/dk are all small enough. This resolves a question of Székely, and generali ..."
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For a set of distances D = {d1,..., dk} a set A is called Davoiding if no pair of points of A is at distance di for some i. We show that the density of A is exponentially small in k provided the ratios d1/d2, d2/d3,..., dk−1/dk are all small enough. This resolves a question of Székely, and generalizes a theorem of FurstenbergKatznelsonWeiss, FalconerMarstrand, and Bourgain. Several more results on Davoiding sets are presented. 1
NILFACTORS OF R mACTIONS AND CONFIGURATIONS IN SETS OF POSITIVE UPPER DENSITY IN R m
, 2005
"... Abstract. We use ergodic theoretic tools to solve a classical problem in geometric Ramsey theory. Let E be a measurable subset of R m, with ¯ D(E)> 0. Let V = {0, v1,..., vk} ⊂ R m. We show that for r large enough, we can find an isometric copy of rV arbitrarily close to E. This is a generalization ..."
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Abstract. We use ergodic theoretic tools to solve a classical problem in geometric Ramsey theory. Let E be a measurable subset of R m, with ¯ D(E)> 0. Let V = {0, v1,..., vk} ⊂ R m. We show that for r large enough, we can find an isometric copy of rV arbitrarily close to E. This is a generalization of a theorem of Furstenberg, Katznelson and Weiss [FuKaW] showing a similar property for m = k = 2. 1.
COVERING THE PLANE BY ROTATIONS OF A LATTICE ARRANGEMENT OF DISKS
, 2006
"... Abstract. Suppose we put an ǫdisk around each lattice point in the plane, and then we rotate this object around the origin for a set Θ of angles. When do we cover the whole plane, except for a neighborhood of the origin? This is the problem we study in this paper. It is very easy to see that if Θ = ..."
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Abstract. Suppose we put an ǫdisk around each lattice point in the plane, and then we rotate this object around the origin for a set Θ of angles. When do we cover the whole plane, except for a neighborhood of the origin? This is the problem we study in this paper. It is very easy to see that if Θ = [0, 2π] then we do indeed cover. The problem becomes more interesting if we try to achieve covering with a small closed set Θ. 1.
Orthogonal systems in vector spaces . . .
, 2008
"... We prove that if a subset of the ddimensional vector space over a finite field is large enough, then it contains many ktuples of mutually orthogonal vectors. ..."
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We prove that if a subset of the ddimensional vector space over a finite field is large enough, then it contains many ktuples of mutually orthogonal vectors.