Let (tk)k=0 be a sequence of real numbers satisfying t0 6= 0 and |tk+1| > (1+ 1/M)|tk| for each k > 0, where M > 1 is a fixed number. We prove that for any sequence of real numbers (ξk)k=0 there is a real number ξ such that ||tkξ−ξk|| > 1/(80M log(28M)) for each k > 0. Here, ||x|| denotes the distance from x ∈ R to the nearest integer. This is a corollary derived from our main theorem which is ...

The aim of present work is to introduce and study lacunary statistical limit and lacunary statistical cluster points for generalized difference sequences of fuzzy numbers. Some inclusion relations among the sets of ordinary limit points, statistical limit points, statistical cluster points, lacunary statistical limit points, and lacunary statistical cluster points for these type of sequences ar...

For any double lacunary sequence θrs = {(kr, ls)} and an admissible ideal I2 ⊆ P(N×N), the aim of present work is to define the concepts of Nθrs(I2)− and Sθrs(I2)−convergence for double sequence of numbers. We also present some inclusion relations between these notions and prove that Sθrs(I2)∩`∞ and S2(I2)∩ `∞ are closed subsets of `∞, the space of all bounded double sequences of numbers.

We give a simple argument proving the lonely runner conjecture in the case where the speed of the runners forms a certain lacunary sequence. This improves an earlier result by Pandey, and is then used to derive that for each number of runners the lonely runner conjecture is true for a set of nonzero measure in a natural probability space.

In this paper we introduce the concepts of double lacunary strongly convergence and double lacunary statistical convergence of double interval numbers. We prove some inclusion relations and study some of their properties. Subjclass [2000] : 40C05,40J05, 46A45.

We study hereditary completeness of systems exponentials on an interval such that the corresponding generating function G is small outside a lacunary sequence intervals Ik. show that, under some technical conditions, exponential system hereditarily complete if and only logarithmic length union these infinite, i.e., $$\sum\nolimits_k {\int_{{I_k}} {{{dx} \over {1 + \left| x \right|}} = \infty } ...