Suppose $S$ is measurable in $[0,1]^2$ and the orthogonal projection of $S$ onto the $x$ and $y$ axes has measure zero. Can the measure of the orthogonal projection of $S$ onto the line $y=x$ be positive?
Each of these measures is the 1 dimensional Lebesgue measure.
If we require just the $x$ axis projection to be zero, the answer is obviously "no" — we can take the entire $y$ axis. The projection has measure $1/2$.
Best Answer
This is essentially the same question as "can the Minkowski sum of two null sets be a non-null set?"
That's because if $S = A \times B$, then the projection of $S$ onto the diagonal is $\{\frac 12(x + y, x + y) : x \in A, y \in B\}$, so its measure will be some multiple of that of $A + B$, and because if $S$ is any set with your property, then $\pi_x(S) \times \pi_y(S)$ is a larger set with the same property.
So there are indeed sets with this property. For example, $C \times C$ where $C$ is the Cantor set is an example.