A density version of the Hales-Jewett theorem

Furstenberg, Harry; Katznelson, Yitzhak

doi:10.1007/bf03041066

Cited by 178 publications

(183 citation statements)

References 12 publications

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“…The most problematic part however turns out to be the base case, where we need to show that the abundance of hyperplanes leads to a cover of most of F n q by q "near-parallel" hyperplanes. For this part we resort to the "density Hales-Jewett theorem" [FK91,Pol09] which says (for our purposes) that for every q and every > 0 there is a c = c q, such that · q c hyperplanes in c dimensions will contain q "near-parallel" ones. (Unfortunately this leads to a horrendous bound on c q, , but fortunately is independent of n and d and so this suffices for Theorem 1.3).…”

Section: Our Main Resultsmentioning

confidence: 99%

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Optimal Testing of Multivariate Polynomials over Small Prime Fields

Haramaty

Shpilka

Sudan

2011

2011 IEEE 52nd Annual Symposium on Foundations of Computer Science

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We consider the problem of testing if a given function f : F n q → F q is close to a n-variate degree d polynomial over the finite field F q of q elements. The natural, low-query, test for this property would be to pick the smallest dimension t = t q,d ≈ d/q such that every function of degree greater than d reveals this feature on some t-dimensional affine subspace of F n q and to test that f when restricted to a random t-dimensional affine subspace is a polynomial of degree at most d on this subspace. Such a test makes only q t queries, independent of n. hold only for fields of prime order). Thus to get a constant probability of detecting functions that were at constant distance from the space of degree d polynomials, the tests made q 2t queries. Kaufman and Ron also noted that when q is prime, then q t queries are necessary. Thus these tests were off by at least a quadratic factor from known lower bounds. It was unclear if the soundness analysis of these tests were tight and this question relates closely to the task of understanding the behavior of the Gowers Norm. This motivated the work of Bhattacharyya et al. [BKS + 10], who gave an optimal analysis for the case of the binary field and showed that the natural test actually rejects functions that were Ω(1)-far from degree d-polynomials with probability at least Ω(1).In this work we give an optimal analysis of this test for all fields showing that the natural test does indeed reject functions that are Ω(1)-far from degree d polynomials with Ω(1)-probability. Our analysis thus shows that this test is optimal (matches known lower bounds) when q is prime. (It is also potentially best possible for all fields.) Our approach extends the proof technique of Bhattacharyya et al., however it has to overcome many technical barriers in the process. The natural extension of their analysis leads to an O(q d ) query complexity, which is worse than that of Kaufman and Ron for all q except 2! The main technical ingredient in our work is a tight analysis of the number of "hyperplanes" (affine subspaces of co-dimension 1) on which the restriction of a degree d polynomial has degree less than d. We show that the number of such hyperplanes is at most O(q t q,d ) -which is tight to within constant factors.

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Section: Our Main Resultsmentioning

confidence: 99%

“…The theorem was first proved by Furstenberg and Katznelson [FK91]. A more recent prove with explicit bounds on the density parameters was obtained in [Pol09].…”

Section: Density Hales-jewett Theoremmentioning

confidence: 99%

Optimal Testing of Multivariate Polynomials over Small Prime Fields

Haramaty

Shpilka

Sudan

2011

2011 IEEE 52nd Annual Symposium on Foundations of Computer Science

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“…Furstenberg and Katznelson ( [FK91]) proved that every stationary process majorizes a strongly stationary one. Actually they established a much more general result using a strong selection theorem.…”

Section: Stationary Processesmentioning

confidence: 99%

The structure of strongly stationary systems

Frantzikinakis

2004

J. Anal. Math.

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Abstract. Motivated by a problem in ergodic Ramsey theory, Furstenberg and Katznelson introduced the notion of strong stationarity, showing that certain recurrence properties hold for arbitrary measure preserving systems if they are valid for strongly stationary ones. We construct some new examples and prove a structure theorem for strongly stationary systems. The building blocks are Bernoulli systems and rotations on nilmanifolds.

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“…This conjecture had to be revised after Alon [Al10] discovered some geometric range spaces of small VC-dimension, in which the ranges are straight lines, rectangles or infinite strips in the plane, and which do not admit ε-nets of size O(1/ε). Alon's construction is based on the density version of the Hales-Jewett theorem [HaJ63], due to Furstenberg and Katznelson [FuK89,FuK91], and recently improved by participants of the Polymath blog project [Po09]. However, Alon's lower bound is only barely superlinear: Ω 1 ε g( 1 ε ) , where g is an extremely slowly growing function, closely related to the inverse Ackermann function.…”

Section: Introductionmentioning

confidence: 99%

Tight lower bounds for the size of epsilon-nets

Pach¹,

Tardos²

2012

J. Amer. Math. Soc.

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According to a well known theorem of Haussler and Welzl (1987), any range space of bounded VC-dimension admits an ε-net of size O 1 ε log 1 ε . Using probabilistic techniques, Pach and Woeginger (1990) showed that there exist range spaces of VCdimension 2, for which the above bound is sharp. The only known range spaces of small VC-dimension, in which the ranges are geometric objects in some Euclidean space and the size of the smallest ε-nets is superlinear in 1 ε , were found by Alon (2010). In his examples, the size of the smallest ε-nets is Ω 1 ε g( 1 ε ) , where g is an extremely slowly growing function, closely related to the inverse Ackermann function.We show that there exist geometrically defined range spaces, already of VCdimension 2, in which the size of the smallest ε-nets is Ω 1 ε log 1 ε . We also construct range spaces induced by axis-parallel rectangles in the plane, in which the size of the smallest ε-nets is Ω 1 ε log log 1 ε . By a theorem of Aronov, Ezra, and Sharir (2010), this bound is tight.

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A density version of the Hales-Jewett theorem

Cited by 178 publications

References 12 publications

Optimal Testing of Multivariate Polynomials over Small Prime Fields

Optimal Testing of Multivariate Polynomials over Small Prime Fields

The structure of strongly stationary systems

Tight lower bounds for the size of epsilon-nets

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