This question came to me while reading the discussion of magic square in the complex plane with equal integrals along every horizontal, vertical and diagonal "magic square in the complex plane with equal integrals along every horizontal, vertical and diagonal"

That question ran into trouble when people pointed out the difficulties involved in just making sense of the question, e.g., what kind of function, what kind of integral, etc. So I made up something halfway between the finite-discrete magic square (that is, the usual kind) and the continuous analogue raised in Question 53352:

What can you say about complex numbers $a_{i,j}$ such that for all $i$ and $j$ the sums $$\sum_{i=-\infty}^{\infty}a_{i,j}=\sum_{j=-\infty}^{\infty}a_{i,j}\lt\infty$$ That is, the sums along vertical and horizontal lines all converge and are all equal.

If the doubly-infinite sums are a worry, just insist that all the numbers be non-negative reals (alternatively, replace the lower limits of summation with zero).

Splitting into real and imaginary parts, we see that we may assume the numbers are real. Other than that, I haven't done much thinking about it.

I realize the number theory tag is not quite appropriate; I'm trying to comply with the direction, "Please try use at least one tag corresponding to an arXiv subject area."