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

Fernando Mário de Oliveira Filho; Frank Vallentin

Journal of the European Mathematical Society (2010)

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

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topde 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 -

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