At the Ithaca meeting in 1946 it was conjectured that it is possible to construct a two-dimensional regular summability matrix $A=\{a_{n,k}\}$ with the property that, for every real sequence $\{s_{k}\}$, the transformed sequence $$t_{n}=\sum_{k=0}^{\infty}a_{n,k}s_{k}$$ possesses at least one limit point in the finite plane. It was also counter-conjectured that, for every regular summability matrix $A$, there exists a single sequence $\{s_{k}\}$ such that the transformed sequence $t_{n}$ tends to infinity monotonically. In 1947 Erdos and Piranian presented answers to these conjectures. The goal of this paper is to present a multidimensional version of the above conjectures. The first conjecture is the following:
A four-dimensional RH-regular summability matrix $A=\{a_{m,n,k,l}\}$ can be constructed with the property that every double sequence $\{s_{k,l}\}$ transformed into the double sequence $$t_{m,n}=\sum_{k,l=0,0}^{\infty,\infty}a_{m,n,k,l}s_{k,l}$$ possesses at least one Pringsheim limit point in the finite plane.
The multidimensional counter-conjecture is the following. For every RH-regular summability matrix $A$ there exists a double sequence $\{s_{k,l}\}$ such that the four-dimensional transformed double sequence $\{t_{m,n}\}$ tends to infinity monotonically Pringsheim sense. This paper established that both multidimensional conjectures are false.}
