# NP-hard $k$-SAT variant with exactly $\ell$ occurrences per variable

For the purpose of this post, let $$k$$-SAT be SAT with exactly $$k$$ literals per clause, as opposed to the more common meaning of at most $$k$$ literals per clause.

With the purpose of proving some problem NP-hard, I'd like to reduce from a $$k$$-SAT variant (which of course should remain NP-hard) in which each variable occurs exactly $$\ell$$ times in the formula.

I've tried to look up 3-SAT with exactly 3 occurrences of a variable, but this paper proves 3-SAT with at most 3 occurrences per variable is trivial, which includes my specific case of exactly 3 occurrences. (In fact, this is the case whenever $$k = \ell$$.)

All hope is not lost though, as in the same paper, it is proven that 3-SAT with at most $$\ell$$ occurrences per variable, with $$\ell>3$$ is NP-hard. I'm looking into the details of the reduction to see if by chance they have proved the specific case of exactly $$\ell$$ occurrences per variable to be NP-hard (which would extend to the general case of at most $$\ell$$ occurrences per variable), but wanted to ask this question here while searching:

Does there exist an NP-hard $$k$$-SAT variant with exactly $$\ell$$ occurrences per variable?

The problem of deciding the satisfiability of 3CNFs in which each variable occurs exactly 5 times is NP-complete (and even NP-hard to approximate), see Uriel Feige, A threshold of $$\ln n$$ for approximating Set Cover.