Fourier analysis, linear programming, and densities of distance avoiding sets in ${ℝ}^n$

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 -