scholarly journals On Kakeya–Nikodym averages, $L^p$-norms and lower bounds for nodal sets of eigenfunctions in higher dimensions

2015 ◽  
Vol 17 (10) ◽  
pp. 2513-2543 ◽  
Author(s):  
Matthew Blair ◽  
Christopher Sogge
Keyword(s):  
Lower Bounds ◽  
Higher Dimensions ◽  
Nodal Sets

scholarly journals Lower Bounds for Nodal Sets of Eigenfunctions

2011 ◽  
Vol 306 (3) ◽  
pp. 777-784 ◽  
Author(s):  
Tobias H. Colding ◽  
William P. Minicozzi
Keyword(s):  
Lower Bounds ◽  
Nodal Sets

scholarly journals Dense Point Sets with Many Halving Lines

2019 ◽  
Vol 64 (3) ◽  
pp. 965-984
Author(s):  
István Kovács ◽  
Géza Tóth
Keyword(s):  
Discrete Comput Geom ◽  
Lower Bounds ◽  
Higher Dimensions ◽  
General Point ◽  
Point Sets ◽  
Planar Point ◽  
Dense Point ◽  
Point Set ◽  
Dense Point Set ◽  
Dense Set

Abstract A planar point set of n points is called $$\gamma $$ γ -dense if the ratio of the largest and smallest distances among the points is at most $$\gamma \sqrt{n}$$ γ n . We construct a dense set of n points in the plane with $$ne^{\Omega ({\sqrt{\log n}})}$$ n e Ω ( log n ) halving lines. This improves the bound $$\Omega (n\log n)$$ Ω ( n log n ) of Edelsbrunner et al. (Discrete Comput Geom 17(3):243–255, 1997). Our construction can be generalized to higher dimensions, for any d we construct a dense point set of n points in $$\mathbb {R}^d$$ R d with $$n^{d-1}e^{\Omega ({\sqrt{\log n}})}$$ n d - 1 e Ω ( log n ) halving hyperplanes. Our lower bounds are asymptotically the same as the best known lower bounds for general point sets.

scholarly journals Dimension estimates for sets of uniformly badly approximable systems of linear forms

2015 ◽  
Vol 11 (07) ◽  
pp. 2037-2054 ◽  
Author(s):  
Ryan Broderick ◽  
Dmitry Kleinbock
Keyword(s):  
Hausdorff Dimension ◽  
Lower Bounds ◽  
Higher Dimensions ◽  
Upper And Lower Bounds ◽  
Homogeneous Dynamics ◽  
Linear Forms ◽  
One Dimensional ◽  
Approximation Constant ◽  
The One ◽  
Badly Approximable

The set of badly approximable m × n matrices is known to have Hausdorff dimension mn. Each such matrix comes with its own approximation constant c, and one can ask for the dimension of the set of badly approximable matrices with approximation constant greater than or equal to some fixed c. In the one-dimensional case, a very precise answer to this question is known. In this note, we obtain upper and lower bounds in higher dimensions. The lower bounds are established via the technique of Schmidt games, while for the upper bound we use homogeneous dynamics methods, namely exponential mixing of flows on the space of lattices.

scholarly journals Lower bounds for interior nodal sets of Steklov eigenfunctions

2016 ◽  
Vol 144 (11) ◽  
pp. 4715-4722 ◽  
Author(s):  
Christopher D. Sogge ◽  
Xing Wang ◽  
Jiuyi Zhu
Keyword(s):  
Lower Bounds ◽  
Nodal Sets ◽  
Steklov Eigenfunctions

scholarly journals Lower bounds on the Hausdorff measure of nodal sets II

2012 ◽  
Vol 19 (6) ◽  
pp. 1361-1364 ◽  
Author(s):  
Christopher D. Sogge ◽  
Steve Zelditch
Keyword(s):  
Lower Bounds ◽  
Hausdorff Measure ◽  
Nodal Sets
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