Geodesic
Fourier analysis, linear programming, and densities of distance avoiding sets in $\mathbb{R}^n$
Journal of the European Mathematical Society, Tome 12 (2010) no. 6, pp. 1417-1428 Cet article a éte moissonné depuis la source EMS Press

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We derive new upper bounds for the densities of measurable sets in 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; . . . ; 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.
DOI : 10.4171/jems/236
Classification : 42-XX, 52-XX, 90-XX, 00-XX
Keywords: Measurable chromatic number, linear programming, autocorrelation function 
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     author = {Fernando M\'ario de Oliveira Filho and Frank Vallentin},
     title = {Fourier analysis, linear programming, and densities of distance avoiding sets in $\mathbb{R}^n$},
     journal = {Journal of the European Mathematical Society},
     pages = {1417--1428},
     year = {2010},
     volume = {12},
     number = {6},
     doi = {10.4171/jems/236},
     url = {http://geodesic.mathdoc.fr/articles/10.4171/jems/236/}
}
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Fernando Mário de Oliveira Filho; Frank Vallentin. Fourier analysis, linear programming, and densities of distance avoiding sets in $\mathbb{R}^n$. Journal of the European Mathematical Society, Tome 12 (2010) no. 6, pp. 1417-1428. doi: 10.4171/jems/236

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