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An incidence theorem in higher dimensions
 Discrete Comput. Geom
"... Abstract. We prove almost tight bounds on incidences between points and kdimensional varieties of bounded degree in R d. Our main tools are the Polynomial Ham Sandwich Theorem and induction on both the dimension and the number of points. ..."
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Abstract. We prove almost tight bounds on incidences between points and kdimensional varieties of bounded degree in R d. Our main tools are the Polynomial Ham Sandwich Theorem and induction on both the dimension and the number of points.
LOWER BOUNDS FOR MEASURABLE CHROMATIC NUMBERS
, 2008
"... The Lovász theta function provides a lower bound for the chromatic number of finite graphs based on the solution of a semidefinite program. In this paper we generalize it so that it gives a lower bound for the measurable chromatic number of distance graphs on compact metric spaces. In particular we ..."
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The Lovász theta function provides a lower bound for the chromatic number of finite graphs based on the solution of a semidefinite program. In this paper we generalize it so that it gives a lower bound for the measurable chromatic number of distance graphs on compact metric spaces. In particular we consider distance graphs on the unit sphere. There we transform the original infinite semidefinite program into an infinite, twovariable linear program which then turns out to be an extremal question about Jacobi polynomials which we solve explicitly in the limit. As an application we derive new lower bounds for the measurable chromatic number of the Euclidean space in dimensions 10,..., 24 and we give a new proof that it grows exponentially with the dimension.
THE DENSITY OF SETS AVOIDING DISTANCE 1 IN EUCLIDEAN SPACE
"... ABSTRACT. We improve by an exponential factor the best known asymptotic upper bound for the density of sets avoiding 1 in Euclidean space. This result is obtained by a combination of an analytic bound that is an analogue of Lovász theta number and of a combinatorial argument involving finite subgra ..."
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ABSTRACT. We improve by an exponential factor the best known asymptotic upper bound for the density of sets avoiding 1 in Euclidean space. This result is obtained by a combination of an analytic bound that is an analogue of Lovász theta number and of a combinatorial argument involving finite subgraphs of the unit distance graph. In turn, we straightforwardly obtain an asymptotic improvement for the measurable chromatic number of Euclidean space. We also tighten previous results for the dimensions between 4 and 24. 1.
SPECTRAL BOUNDS FOR THE INDEPENDENCE RATIO AND THE CHROMATIC NUMBER OF AN OPERATOR
, 2013
"... We define the independence ratio and the chromatic number for bounded, selfadjoint operators on an L2space by extending the definitions for the adjacency matrix of finite graphs. In analogy to the Hoffman bounds for finite graphs, we give bounds for these parameters in terms of the numerical range ..."
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We define the independence ratio and the chromatic number for bounded, selfadjoint operators on an L2space by extending the definitions for the adjacency matrix of finite graphs. In analogy to the Hoffman bounds for finite graphs, we give bounds for these parameters in terms of the numerical range of the operator. This provides a theoretical framework in which many packing and coloring problems for finite and infinite graphs can be conveniently studied with the help of harmonic analysis and convex optimization. The theory is applied to infinite geometric graphs on Euclidean space and on the unit sphere.
FOURIER ANALYSIS, LINEAR PROGRAMMING, AND DENSITIES OF DISTANCE AVOIDING SETS IN R n
, 808
"... Abstract. 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 ..."
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Abstract. 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. 1.
Russian Math. Surveys 56:1 103–139 c○2001 RAS(DoM) and LMS Uspekhi Mat. Nauk 56:1 107–146 DOI 10.1070/RM2001v056n01ABEH000358
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DistanceAvoiding Sets in the Plane
, 2014
"... Fix a real number 0. Let = {1 } if 6 = 1; otherwise = {1} may simply be written as 1. A subset ⊆ R is said to avoid if k − k ∈ for all ∈ . For example, the union of open balls of radius 12 with centers in (2Z) avoids the distance 1. If instead the balls have centers in (3Z), th ..."
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Fix a real number 0. Let = {1 } if 6 = 1; otherwise = {1} may simply be written as 1. A subset ⊆ R is said to avoid if k − k ∈ for all ∈ . For example, the union of open balls of radius 12 with centers in (2Z) avoids the distance 1. If instead the balls have centers in (3Z), then their union avoids {1 2}. It is natural to ask about the “largest possible ” that avoids . Let denote the ball of radius with center 0. Assuming is Lebesgue measurable, its density () = limsup quantifies the asymptotic proportion of R occupied by . We wish to know (R) = sup {() : is measurable and avoids } The shortage of information regarding (R) is surprising. Until further notice, let = 2 and = 1 for simplicity [1, 2, 3]. On the one hand, the number of Z2 points within is ∼ 2 [4], hence the number of (2Z)2 points within is ∼ (4)2. Each open disk in our example has area 4 and has area 2, thus 1(R2) ≥ 16 ≈ 0196. It turns out we can do better by arranging the disks with centers according to an equilateral triangle lattice, giving 1(R2) ≥ ¡8√3 ¢ ≈ 0227. An additional improvement (replacing six portions of each circular circumference by linear segments) gives 1(R2) ≥ 0229365. This is the best lower bound currently known [5, 6]. On the other hand, a configuration called the Moser spindle implies that1(R2) ≤ 27 ≈ 0286 [7, 8]. Székely [9, 10] improved the upper bound to 1243 ≈ 0279. The best result currently known is 1(R2) ≤ 0268412 via linear programming techniques [11]. Erdős ’ conjecture that 1(R2) 14 seems out of reach. Sets avoiding 1 have been studied by combinatorialists because of their association with the measurable chromatic number of the plane. What is the minimum number of colors (R2) required to color all points of R2 so that any two points at distance 1 receive distinct colors and so that points receiving the same color form Lebesgue measurable sets? It is known only that 5 ≤ (R2) ≤ 7 [12].