Here is a summary for the $\mathbb{R}^2$ situation.

Upper limit: no improvement over the 3.8 known on $\mathbb{Z}^2$.

To create specific examples I modified the $\mathbb{Z}^2$ ones by uniformely
spreading each value on the lattice over a 1x1 square:

3x1:  ratio = 3-3/5 (vs. 3 on integers, 7x5 grid)  
4x1:  ratio = 3-1/5 (vs. 3.5 on integers, 13x16 grid)  
5x1:  ratio = 3-1/7 (vs. 25/7 on integers, 25x29 grid)  
6x1:  ratio = 3-1/23 (vs. 85/23 on integers, 31x36 grid)  
8x1:  ratio = 3-1/35 (vs. 26/7 on integers, 39x46 grid)  
7x3:  ratio = 3-1/75 (vs. 56/15 on integers, 59x57 grid)  
11x1: ratio = 3-1/135 (vs. 101/27 on integers, 137x63 grid)

And the surprises are  
1) that in all cases the highest sum on a square is 1.25 that of the integral case (which is the worst case scenario, but I see no obvious reason it should always be that way), and  
2) we seem to be approaching 3, suggesting that for the integers perhaps the upper limit could be 3.75 and not 3.8 - but this is of course very highly speculative...