Suppose that K ⊆ R, d ≥ 2, is a 0-symmetric convex body which defines the usual norm ‖x‖K = sup {t ≥ 0 : x / ∈ tK} on R. Let also A ⊆ R be a measurable set of positive upper density ρ. We show that if the body K is not a polytope, or if it is a polytope with many faces (depending on ρ), then the distance set DK(A) = {‖x− y‖K : x, y ∈ A} contains all points t ≥ t0 for some positive number t0. Th...

We use mixed norm estimates for the spherical averaging operator to obtain some results concerning pinned distance sets.

A subset X in the d-dimensional Euclidean space is called a k-distance set if there are exactly k distinct distances between two distinct points in X and a subset X is called a locally k-distance set if for any point x in X , there are at most k distinct distances between x and other points in X . Delsarte, Goethals, and Seidel gave the Fisher type upper bound for the cardinalities of k-distanc...

Fix a real number d > 0. Let D = {1, d} if d 6= 1; otherwise D = {1} may simply be written as 1. A subset S ⊆ R is said to avoid D if kx− yk / ∈ D for all x, y ∈ S. For example, the union of open balls of radius 1/2 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” S that avoid...

In this paper we discuss the Erdős-Falconer distance problem. The classical Erdős distance problem in R, d 2, asks for the smallest possible size of (E) = {|x y| : x, y 2 E} with E ⇢ R a finite set. An analogous problem is the Falconer distance problem which asks how large does the Hausdor↵ dimension of a compact set E ⇢ R, d 2, needs to be to ensure that the Lebesgue measure of (E), defined as...

Rotation distance measures the difference in shape between binary trees of the same size by counting the minimum number of rotations needed to transform one tree to the other. We describe several types of rotation distance where restrictions are put on the locations where rotations are permitted, and provide upper bounds on distances between trees with a fixed number of nodes with respect to se...

A planar point set X in the Euclidean plane is called a k-distance set if there are exactly k distances between two distinct points in X. An interesting problem is to find the largest possible cardinality of k-distance sets. This problem was introduced by Erdős and Fishburn (1996). Maximum planar sets that determine k distances for k less than 5 has been identified. The 6-distance conjecture of...

the aim of this study was to study the effects of two fitness training programs with long and short sets with the same intensity and volume on physical fitness and performance factors in rugby elite players. 27 rugby players from rugby club of khorasan razavi gas industrial cooperative (age 24.29+3.172 yr and height 182.22+5.652 cm) participated voluntarily and were divided randomly into three ...

Rotation distance quantifies the difference in shape between two rooted binary trees of the same size by counting the minimum number of elementary changes needed to transform one tree to the other. We describe several types of rotation distance, and provide upper bounds on distances between trees with a fixed number of nodes with respect to each type. These bounds are obtained by relating each ...

Rotation distance quantifies the difference in shape between two rooted binary trees of the same size by counting the minimum number of elementary changes needed to transform one tree to the other. We describe several types of rotation distance, and provide upper bounds on distances between trees with a fixed number of nodes with respect to each type. These bounds are obtained by relating each ...