Do you want to publish a course? Click here

Measurable sets with excluded distances

52   0   0.0 ( 0 )
 Added by Boris Bukh
 Publication date 2007
  fields
and research's language is English
 Authors Boris Bukh




Ask ChatGPT about the research

For a set of distances D={d_1,...,d_k} a set A is called D-avoiding if no pair of points of A is at distance d_i for some i. We show that the density of A is exponentially small in k provided the ratios d_1/d_2, d_2/d_3, ..., d_{k-1}/d_k are all small enough. This resolves a question of Szekely, and generalizes a theorem of Furstenberg-Katznelson-Weiss, Falconer-Marstrand, and Bourgain. Several more results on D-avoiding sets are presented.



rate research

Read More

294 - Chengfei Xie , Gennian Ge 2020
A Nikodym set $mathcal{N}subseteq(mathbb{Z}/(Nmathbb{Z}))^n$ is a set containing $Lsetminus{x}$ for every $xin(mathbb{Z}/(Nmathbb{Z}))^n$, where $L$ is a line passing through $x$. We prove that if $N$ is square-free, then the size of every Nikodym set is at least $c_nN^{n-o(1)}$, where $c_n$ only depends on $n$. This result is an extension of the result in the finite field case.
120 - Moaaz AlQady 2020
We study ErdH oss distinct distances problem under $ell_p$ metrics with integer $p$. We improve the current best bound for this problem from $Omega(n^{4/5})$ to $Omega(n^{6/7-epsilon})$, for any $epsilon>0$. We also characterize the sets that span an asymptotically minimal number of distinct distances under the $ell_1$ and $ell_infty$ metrics.
Frame matroids and lifted-graphic matroids are two distinct minor-closed classes of matroids, each of which generalises the class of graphic matroids. The class of quasi-graphic matroids, recently introduced by Geelen, Gerards, and Whittle, simultaneously generalises both the classes of frame and lifted-graphic matroids. Let $mathcal{M}$ be one of these three classes, and let $r$ be a positive integer. We show that $mathcal{M}$ has only a finite number of excluded minors of rank $r$.
193 - Tao Feng , Sihuang Hu , Shuxing Li 2013
The known families of difference sets can be subdivided into three classes: difference sets with Singer parameters, cyclotomic difference sets, and difference sets with gcd$(v,n)>1$. It is remarkable that all the known difference sets with gcd$(v,n)>1$ have the so-called character divisibility property. In 1997, Jungnickel and Schmidt posed the problem of constructing difference sets with gcd$(v,n)>1$ that do not satisfy this property. In an attempt to attack this problem, we use difference sets with three nontrivial character values as candidates, and get some necessary conditions.
comments
Fetching comments Fetching comments
Sign in to be able to follow your search criteria
mircosoft-partner

هل ترغب بارسال اشعارات عن اخر التحديثات في شمرا-اكاديميا