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

Abstract

Abstract. 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.

On some problems of Euclidean Ramsey theory

“…The proof follows the arguments from [9] as well as the approach (and the notation) from the previous section.…”

“…The proof uses simple Fourier analysis in spirit of paper [9] and hugely relies on the fact that we have deal just with two colors. Also we consider a model situation of the plane over the prime finite field F p × F p and prove (a slightly stronger) analog of Theorem 1, see section 2.…”

Spectral bounds for the independence ratio and the chromatic number of an operator

“…There are several recent results which can be conveniently interpreted as examples of our theory: In Section 3 we compute spectral bounds for graphs defined on the Euclidean space which are invariant under translations. Thereby we can recover results from Oliveira and Vallentin [27], Kolountzakis [23], and Steinhardt [32]. In Section 4 we determine spectral bounds for distance graphs defined on the unit sphere and generalize results from Bachoc, Nebe, Oliveira, and Vallentin [6].…”

“…and an extension of the technique of Oliveira and Vallentin [27] who gave an upper bound for the measurable chromatic number of graphs of the form…”

Bloch-sphere colorings and Bell inequalities

“…[21,22] From the CHSH inequality, (4), we obtain after averaging over random rotations of the Bloch sphere that |3C x (θ) − C x (0)| ≤ 2. Then, the result follows because, as shown in the main text, we have C x (0) = −1 + 2γ.…”