We call a finitely generated group lacunary hyperbolic if one of its asymptotic cones is an R-tree. We characterize lacunary hyperbolic groups as direct limits of Gromov hyperbolic groups satisfying certain restrictions on the hyperbolicity constants and injectivity radii. Using central extensions of lacunary hyperbolic groups, we solve a problem of Gromov by constructing a group whose asymptot...

This paper presents the following definition which is a natural combination of the definition for asymptotically equivalent, statistically limit and lacunary sequences. Let θ be a lacunary sequence; the two nonnegative sequences x = (xk) and y = (yk) are said to be asymptotically ∆ m lacunary statistical(defined in [2]) equivalent of multiple L provided that for every > 0, lim r 1 hr { the numb...

By a classical result of Weyl (1916), for any increasing sequence (nk) of positive integers, (nkx) is uniformly distributed mod 1 for almost all x. The precise asymptotics of the discrepancy of this sequence is known only in a few cases, e.g. for nk = k (Khinchin (1924)) and for lacunary (nk) (Philipp (1975)). In this paper we extend Philipp’s result to lacunary sequences with multidimensional ...

In this article we introduce and study the notions of double -lacunary strongly summable, double  Cesàro strongly summable, double  statistically convergent and double  -lacunary statistically convergent sequence of fuzzy numbers. Consequently we construct the spaces   2 F L  ,   2 F   ,   2 F S  and   2 F S  respectively and investigate the relationship among these spaces. F...

In the present paper, we study the geometric discrepancy with respect to families of rotated rectangles. The well-known extremal cases are the axis-parallel rectangles (logarithmic discrepancy) and rectangles rotated in all possible directions (polynomial discrepancy). We study several intermediate situations: lacunary sequences of directions, lacunary sets of finite order, and sets with small ...

We call a finitely generated group lacunary hyperbolic if one of its asymptotic cones is an R-tree. We characterize lacunary hyperbolic groups as direct limits of Gromov hyperbolic groups satisfying certain restrictions on the hyperbolicity constants and injectivity radii. Using central extensions of lacunary hyperbolic groups, we solve a problem of Gromov by constructing a group whose asymptot...

This paper presents the following definition which is a natural combination of the definition for asymptotically equivalent of order α, where 0 < α 6 1, I-statistically limit, and I-lacunary statistical convergence for sequences of sets. Let (X, ρ) be a metric space and θ be a lacunary sequence. For any non-empty closed subsets Ak, Bk ⊆ X such that d(x,Ak) > 0 and d(x,Bk) > 0 for each x ∈ X, we...

In 1987 Paul Erdős asked me if the Cayley graph defined on Z by a lacunary sequence has necessarily a finite chromatic number. Below is my answer, delivered to him on the spot but neverpublished, and some additional remarks. The key is the interpretation of the question in terms of return times of dynamical systems. 1 The Cayley graph ZΛ 1.1 Let Λ ⊂ N. By definition, the Cayley graph ZΛ is the ...

In this paper, we generalize the concept of I statistical convergence, which is a recently introduced summability method, with an in…nite matrix of complex numbers and a modulus function. Lacunary sequences will also be included in our de…nitions. The name of our new method will be AI -lacunary statistical convergence with respect to a sequence of modulus functions. We also de…ne strongly AI -l...