7,291 research outputs found
Density estimates of 1-avoiding sets via higher order correlations
We improve the best known upper bound on the density of a planar measurable
set A containing no two points at unit distance to 0.25442. We use a
combination of Fourier analytic and linear programming methods to obtain the
result. The estimate is achieved by means of obtaining new linear constraints
on the autocorrelation function of A utilizing triple-order correlations in A,
a concept that has not been previously studied.Comment: 10 pages, 2 figure
Quadrisecants give new lower bounds for the ropelength of a knot
Using the existence of a special quadrisecant line, we show the ropelength of
any nontrivial knot is at least 15.66. This improves the previously known lower
bound of 12. Numerical experiments have found a trefoil with ropelength less
than 16.372, so our new bounds are quite sharp.Comment: v3 is the version published by Geometry & Topology on 25 February
200
The density of planar sets avoiding unit distances
By improving upon previous estimates on a problem posed by L. Moser, we prove
a conjecture of Erd\H{o}s that the density of any measurable planar set
avoiding unit distances cannot exceed . Our argument implies the upper
bound of .Comment: 24 pages, 6 figures. Final version, to appear in Mathematical
Programmin
On the density of sets avoiding parallelohedron distance 1
The maximal density of a measurable subset of R^n avoiding Euclidean
distance1 is unknown except in the trivial case of dimension 1. In this paper,
we consider thecase of a distance associated to a polytope that tiles space,
where it is likely that the setsavoiding distance 1 are of maximal density
2^-n, as conjectured by Bachoc and Robins. We prove that this is true for n =
2, and for the Vorono\"i regions of the lattices An, n >= 2
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