skip to main content
US FlagAn official website of the United States government
dot gov icon
Official websites use .gov
A .gov website belongs to an official government organization in the United States.
https lock icon
Secure .gov websites use HTTPS
A lock ( lock ) or https:// means you've safely connected to the .gov website. Share sensitive information only on official, secure websites.

Explore Research Products in the PAR
It may take a few hours for recently added research products to appear in PAR search results.


  • Home
  • Search Results
  • Page 1 of 1

Search for: All records

Award ID contains: 2154356

Note: When clicking on a Digital Object Identifier (DOI) number, you will be taken to an external site maintained by the publisher. Some full text articles may not yet be available without a charge during the embargo (administrative interval).
What is a DOI Number?

Some links on this page may take you to non-federal websites. Their policies may differ from this site.

  1. ; ; (, The Journal of Geometric Analysis)
    Abstract For every $$\beta \in (0,\infty )$$ β ∈ ( 0 , ∞ ) , $$\beta \ne 1$$ β ≠ 1 , we prove that a positive measure subset A of the unit square contains a point $$(x_0,y_0)$$ ( x 0 , y 0 ) such that A nontrivially intersects curves $$y-y_0 = a (x-x_0)^\beta $$ y - y 0 = a ( x - x 0 ) β for a whole interval $$I\subseteq (0,\infty )$$ I ⊆ ( 0 , ∞ ) of parameters $$a\in I$$ a ∈ I . A classical Nikodym set counterexample prevents one to take $$\beta =1$$ β = 1 , which is the case of straight lines. Moreover, for a planar set A of positive density, we show that the interval I can be arbitrarily large on the logarithmic scale. These results can be thought of as Bourgain-style large-set variants of a recent continuous-parameter Sárközy-type theorem by Kuca, Orponen, and Sahlsten. 
    more » « less
    Full Text Available 
  2. ; ; (, Mathematische Zeitschrift)
    Full Text Available