Fourier analysis, linear programming, and densities of distance avoiding sets in R^n
By Fernando Mário de Oliveira Filho, Frank Vallentin. 2008
59 citations recorded by the source corpus.
In this paper 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 new, short proof of a recent result of Bukh which in turn generalizes theorems of Furstenberg, Katznelson, Weiss and Bourgain and Falconer about sets avoiding many distances.
Subfields: Applied Mathematics, Discrete Mathematics and Combinatorics, Statistics and Probability
View Paper: https://www.semanticscholar.org/paper/de3ed28ee95a864cfbd921d582f367c4d5252aa3
How researchers see this paper
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- Frank Vallentin
- C. Bachoc
- Gabriele Nebe
- Fernando Mário Oliveira Filho
- Pierre de la Harpe
Citations in the corpus
- Lower Bounds for Measurable Chromatic Numbers (2008, background): “or n = 10,...,24. We give them in Table 8.1. For n = 2,...,8 our bounds are worse than the existing ones and for n = 9 our bound is 35 which is also the best known one. In fact Oliveira and Vallentin [16] show, by different methods, that the above bound is actually a bound for χm(Rn−1). This then gi”