792 research outputs found
Sets which are not tube null and intersection properties of random measures
We show that in there are purely unrectifiable sets of
Hausdorff (and even box counting) dimension which are not tube null,
settling a question of Carbery, Soria and Vargas, and improving a number of
results by the same authors and by Carbery. Our method extends also to "convex
tube null sets", establishing a contrast with a theorem of Alberti,
Cs\"{o}rnyei and Preiss on Lipschitz-null sets. The sets we construct are
random, and the proofs depend on intersection properties of certain random
fractal measures with curves.Comment: 24 pages. Referees comments incorporated. JLMS to appea
Restriction and decay for flat hypersurfaces
In the first part we consider restriction theorems for hypersurfaces [Gamma] in Rn, with the affine curvature [fòrmula] introduced as a mitigating factor. Sjolin, [19], showed that there is a universal restriction theorem for all convex curves in R2. We show that in dimensions greater than two there is no analogous universal restriction theorem for hypersurfaces with non-negative curvature. In the second part we discuss decay estimates for the Fourier transform of the density [fòrmula] supported on the surface and investigate the relationship between restriction and decay in this setting. It is well-known that restriction theorems follow from appropriate decay estimates; one would like to know whether restriction and decay are, in fact, equivalent. We show that this is not the case in two dimensions. We also go some way towards a classification of those curves/surfaces for which decay holds by giving some sufficient conditions and some necessary conditions for decay
Counting joints in vector spaces over arbitrary fields
We give a proof of the "folklore" theorem that the
Kaplan--Sharir--Shustin/Quilodr\'an result on counting joints associated to a
family of lines holds in vector spaces over arbitrary fields, not just the
reals. We also discuss a distributional estimate on the multiplicities of the
joints in the case that the family of lines is sufficiently generic.Comment: Not intended for publication. References added and other minor edits
in this versio
The endpoint multilinear Kakeya theorem via the Borsuk--Ulam theorem
We give an essentially self-contained proof of Guth's recent endpoint
multilinear Kakeya theorem which avoids the use of somewhat sophisticated
algebraic topology, and which instead appeals to the Borsuk-Ulam theorem
On the Multilinear Restriction and Kakeya conjectures
We prove -linear analogues of the classical restriction and Kakeya
conjectures in . Our approach involves obtaining monotonicity formulae
pertaining to a certain evolution of families of gaussians, closely related to
heat flow. We conclude by giving some applications to the corresponding
variable-coefficient problems and the so-called "joints" problem, as well as
presenting some -linear analogues for .Comment: 38 pages, no figures, submitte
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