74 research outputs found
An improved bound on the number of point-surface incidences in three dimensions
We show that points and smooth algebraic surfaces of bounded degree
in satisfying suitable nondegeneracy conditions can have at most
incidences, provided that any
collection of points have at most O(1) surfaces passing through all of
them, for some . In the case where the surfaces are spheres and no
three spheres meet in a common circle, this implies there are point-sphere incidences. This is a slight improvement over the previous
bound of for an (explicit) very
slowly growing function. We obtain this bound by using the discrete polynomial
ham sandwich theorem to cut into open cells adapted to the set
of points, and within each cell of the decomposition we apply a Turan-type
theorem to obtain crude control on the number of point-surface incidences. We
then perform a second polynomial ham sandwich decomposition on the irreducible
components of the variety defined by the first decomposition. As an
application, we obtain a new bound on the maximum number of unit distances
amongst points in .Comment: 17 pages, revised based on referee comment
Spectral gaps, additive energy, and a fractal uncertainty principle
We obtain an essential spectral gap for -dimensional convex co-compact
hyperbolic manifolds with the dimension of the limit set close to
. The size of the gap is expressed using the additive energy of
stereographic projections of the limit set. This additive energy can in turn be
estimated in terms of the constants in Ahlfors-David regularity of the limit
set. Our proofs use new microlocal methods, in particular a notion of a fractal
uncertainty principle.Comment: 85 pages, 10 figures. To appear in GAF
A Kakeya maximal function estimate in four dimensions using planebrushes
We obtain an improved Kakeya maximal function estimate in
using a new geometric argument called the planebrush. A planebrush is a higher
dimensional analogue of Wolff's hairbrush, which gives effective control on the
size of Besicovitch sets when the lines through a typical point concentrate
into a plane. When Besicovitch sets do not have this property, the existing
trilinear estimates of Guth-Zahl can be used to bound the size of a Besicovitch
set. In particular, we establish a maximal function estimate in
at dimension . As a consequence, every Besicovitch set in
must have Hausdorff dimension at least .Comment: 40 pages 2 figures. v2: revised based on referee's comments. In v1,
the Nikishin-Pisier-Stein factorization theorem was stated (and used)
incorrectly. This version corrects the problem by introducing several new
arguments. The new argument leads to a Kakeya maximal function estimate at
dimension 3.059, which is slightly worse than the previously claimed exponent
3.085
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