In this report, we first review some algebraic and topological structures relevant to some difference sequence spaces which were constructed in comparison with the classical spaces 0 c , c and ∞ l .Then we review Cesàro and lacunary summability methods for difference sequences. Lastly, we review algebraic duals for several different kinds of difference sequence spaces.

Let ( a n ) = 1 ? $(a_{n})_{n=1}^{\infty }$ be lacunary sequence of positive real numbers. Rudnick and Technau showed that for almost all ? ? R $\alpha \in \mathbb {R}$ , the pair correlation $(\alpha a_{n})_{n=1}^{\infty mod is Poissonian. We show higher correlations hence nearest-neighbour spacing distribution are Poissonian as well, thereby extending result Zaharescu to real-valued sequences.

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 prove Strassen’s law of the iterated logarithm for sums ∑N k=1 f(nkx), where f is a smooth periodic function on the real line and (nk)k≥1 is an increasing random sequence. Our results show that classical results of the theory of lacunary series remain valid for sequences with random gaps, even in the nonharmonic case and if the Hadamard gap condition fails.

*Correspondence: [email protected] 1Department of Mathematics, Rajiv Gandhi University, Rono Hills, Doimukh, Arunachal Pradesh 791112, India Full list of author information is available at the end of the article Abstract An ideal I is a family of subsets of positive integers N×N which is closed under taking finite unions and subsets of its elements. In this paper, we present some definitions w...

A primary example is to take an integer g ≥ 2 and set a(x) = g. As is true for any increasing sequence of integers, for almost every α the fractional parts αa(x) are uniformly distributed modulo 1. Moreover, for lacunary sequences, it has long been known that the fractional parts of αa(x) have strong randomness properties. For instance, the exponential sums 1 √ N ∑ x≤N cos(2παa(x)) have a Gauss...