Two DFA intersection emptiness connections to SETH & L vs P
The "inverse" is almost the same as


  SAT is solvable in $O(2^{(1-\epsilon)n})$ time implies the intersection problem is solvable in
  $O(n^{2-\epsilon})$ time.


To show this, it seems that you would need to provide a reduction from an intersection problem instance of size $n$ to a SAT instance of size $2\cdot log_2(n)$.

This kind of reduction would be very interesting because it would take an instance of one problem to a much smaller instance of another problem.  As a result, this reduction would not be one-to-one.  Actually, there would be exponentially many inputs that map to the same output.

If you know of any interesting reductions of this form, please provide a reference.  Thank you.  :)