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Green used an arithmetic analogue of Szemerédi's celebrated regularity lemma to prove the following strengthening of Roth's theorem in vector spaces. For every α>0, β<α3, and prime number p, there is a least positive integer n_p(α,β) such that if n≥n_p(α,β), then for every subset of 𝔽np of density at least α there is a nonzero d for which the density of three-term arithmetic progressions with common difference d is at least β. We determine for p≥19 the tower height of n_p(α,β) up to an absolute constant factor and an additive term depending only on p. In particular, if we want half the random bound (so β=α^{3}/2), then the dimension n required is a tower of twos of height Θ((log p)loglog(1/α)). It turns out that the tower height in general takes on a different form in several different regions of α and β, and different arguments are used both in the upper and lower bounds to handle these cases.
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DISTORTION IN THE FINITE DETERMINATION RESULT FOR EMBEDDINGS OF LOCALLY FINITE METRIC SPACES INTO BANACH SPACES
Abstract Given a Banach space X and a real number α ≥ 1, we write: (1) D ( X ) ≤ α if, for any locally finite metric space A , all finite subsets of which admit bilipschitz embeddings into X with distortions ≤ C , the space A itself admits a bilipschitz embedding into X with distortion ≤ α ⋅ C ; (2) D ( X ) = α + if, for every ϵ > 0, the condition D ( X ) ≤ α + ϵ holds, while D ( X ) ≤ α does not; (3) D ( X ) ≤ α + if D ( X ) = α + or D ( X ) ≤ α. It is known that D ( X ) is bounded by a universal constant, but the available estimates for this constant are rather large. The following results have been proved in this work: (1) D ((⊕ n =1 ∞ X n ) p ) ≤ 1 + for every nested family of finite-dimensional Banach spaces { X n } n =1 ∞ and every 1 ≤ p ≤ ∞. (2) D ((⊕ n =1 ∞ ℓ ∞ n ) p ) = 1 + for 1 < p < ∞. (3) D ( X ) ≤ 4 + for every Banach space X with no nontrivial cotype. Statement (3) is a strengthening of the Baudier–Lancien result (2008).
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- Award ID(s):
- 1700176
- PAR ID:
- 10089970
- Date Published:
- Journal Name:
- Glasgow Mathematical Journal
- Volume:
- 61
- Issue:
- 1
- ISSN:
- 0017-0895
- Page Range / eLocation ID:
- 33 to 47
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1017/S0017089518000022
