*(English)*Zbl 1106.40002

*A. R. Freedman*, *J. J. Sember* and *M. Raphael* [Proc. Lond. Math. Soc., III. Ser. 37, 508–520 (1978; Zbl 0424.40008)] presented a definition for lacunary refinement as follows: $\rho =\left\{{\overline{k}}_{r}\right\}$ is called a lacunary refinement of the lacunary sequence $\theta =\left\{{k}_{r}\right\}$ if $\left\{{k}_{r}\right\}\subseteq \left\{{\overline{k}}_{r}\right\}$. They use this definition to present a one-side inclusion with respect to the refined and nonrefined sequence. *J. Li* [Int. J. Math. Math. Sci. 23, 175–180 (2000; Zbl 0952.40001)] presented the other side of the inclusion.

In this paper, the authors present a multidimensional analogue to the notion of refinement of lacunary sequences, and use this definition to present both sides of the above inclusion. In addition, the authors present a notion of double lacunary statistically Cauchy convergence and use this definition to establish that it is equivalent to the ${S}_{\theta r,s}$-$P$-convergence. For details, we refer the reader to the paper.