Fourier analysis, linear programming, and densities of distance avoiding sets in n

Fernando Mário de Oliveira Filho; Frank Vallentin

Journal of the European Mathematical Society (2010)

  • Volume: 012, Issue: 6, page 1417-1428
  • ISSN: 1435-9855

Abstract

top
We derive new upper bounds for the densities of measurable sets in n which avoid a finite set of prescribed distances. The new bounds come from the solution of a linear programming problem. We apply this method to obtain new upper bounds for measurable sets which avoid the unit distance in dimensions 2 , , 24 . This gives new lower bounds for the measurable chromatic number in dimensions 3 , , 24 . We apply it to get a short proof of a variant of a recent result of Bukh which in turn generalizes theorems of Furstenberg, Katznelson,Weiss, Bourgain and Falconer about sets avoiding many distances.

How to cite

top

de Oliveira Filho, Fernando Mário, and Vallentin, Frank. "Fourier analysis, linear programming, and densities of distance avoiding sets in $\mathbb {R}^n$." Journal of the European Mathematical Society 012.6 (2010): 1417-1428. <http://eudml.org/doc/277796>.

@article{deOliveiraFilho2010,
abstract = {We derive new upper bounds for the densities of measurable sets in $\mathbb \{R\}^n$ which avoid a finite set of prescribed distances. The new bounds come from the solution of a linear programming problem. We apply this method to obtain new upper bounds for measurable sets which avoid the unit distance in dimensions $2,\dots ,24$. This gives new lower bounds for the measurable chromatic number in dimensions $3,\dots ,24$. We apply it to get a short proof of a variant of a recent result of Bukh which in turn generalizes theorems of Furstenberg, Katznelson,Weiss, Bourgain and Falconer about sets avoiding many distances.},
author = {de Oliveira Filho, Fernando Mário, Vallentin, Frank},
journal = {Journal of the European Mathematical Society},
keywords = {measurable chromatic number; linear programming; autocorrelation function; measurable chromatic number; autocorrelation function},
language = {eng},
number = {6},
pages = {1417-1428},
publisher = {European Mathematical Society Publishing House},
title = {Fourier analysis, linear programming, and densities of distance avoiding sets in $\mathbb \{R\}^n$},
url = {http://eudml.org/doc/277796},
volume = {012},
year = {2010},
}

TY - JOUR
AU - de Oliveira Filho, Fernando Mário
AU - Vallentin, Frank
TI - Fourier analysis, linear programming, and densities of distance avoiding sets in $\mathbb {R}^n$
JO - Journal of the European Mathematical Society
PY - 2010
PB - European Mathematical Society Publishing House
VL - 012
IS - 6
SP - 1417
EP - 1428
AB - We derive new upper bounds for the densities of measurable sets in $\mathbb {R}^n$ which avoid a finite set of prescribed distances. The new bounds come from the solution of a linear programming problem. We apply this method to obtain new upper bounds for measurable sets which avoid the unit distance in dimensions $2,\dots ,24$. This gives new lower bounds for the measurable chromatic number in dimensions $3,\dots ,24$. We apply it to get a short proof of a variant of a recent result of Bukh which in turn generalizes theorems of Furstenberg, Katznelson,Weiss, Bourgain and Falconer about sets avoiding many distances.
LA - eng
KW - measurable chromatic number; linear programming; autocorrelation function; measurable chromatic number; autocorrelation function
UR - http://eudml.org/doc/277796
ER -

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.