Sets of large dimension not containing polynomial configurations

Details Sets of large dimension not containing polynomial configurations Sets of large dimension not containing polynomial configurations

2012-01-02

arxiv.org/abs/1201.0548v1

Mathematics

Classical Analysis and ODEs

Scientific paper

The main result of this paper is the following. Given countably many multivariate polynomials with rational coefficients and maximum degree $d$, we construct a compact set $E\subset \R^n$ of Hausdorff dimension $n/d$ which does not contain finite point configurations corresponding to the zero sets of the given polynomials. Given a set $E\subset \R^n$, we study the angles determined by three points of $E$. The main result implies the existence of a compact set in $\R^n$ of Hausdorff dimension $n/2$ which does not contain the angle $\pi/2$. (This is known to be sharp if $n$ is even.) We show that there is a compact set of Hausdorff dimension $n/8$ which does not contain an angle in any given countable set. We also construct a compact set $E\subset \R^n$ of Hausdorff dimension $n/6$ for which the set of angles determined by $E$ is Lebesgue null. In the other direction, we present a result that every set of sufficiently large dimension contains an angle $\epsilon$ close to any given angle.

Affiliated with

Also associated with

No associations

Rating

Sets of large dimension not containing polynomial configurations does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Sets of large dimension not containing polynomial configurations, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Sets of large dimension not containing polynomial configurations will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-183319