Given a sequence $S$ of natural numbers, write ${\bf Gap}(S)$ for the set of differences between consecutive terms. (So $|{\bf Gap}(S)|=1$ precisely for arithmetic progressions, hence the connection to Erdös-Newman-Misrky.)

Question 1: Can one partition the natural numbers into finitely many sequences $S_i$ with all the ${\bf Gap}(S_i)$ collectively disjoint and with no ${\bf Gap}(S_i)$ containing 1?

Question 2: If so, can one do it with $|{\bf Gap}(S_i)|<\infty$ for all $i$?

Question 3: If Question 2 comes up negative, can one prove this in something like the usual way, via generating functions and their poles? (I believe that a counter-example plus the pigeon-hole principle would yield a periodic counter-example.)

Given a sequence $S$ of natural numbers, write ${\bf Gap}(S)$ for the set of differences between consecutive terms. (So $|{\bf Gap}(S)|=1$ precisely for arithmetic progressions, hence the connection to Erdős–Newman–Mirsky.)

Question 1: Can one partition the natural numbers into finitely many sequences $S_i$ with all the ${\bf Gap}(S_i)$ collectively disjoint and with no ${\bf Gap}(S_i)$ containing 1?

Question 2: If so, can one do it with $|{\bf Gap}(S_i)|<\infty$ for all $i$?

Question 3: If Question 2 comes up negative, can one prove this in something like the usual way, via generating functions and their poles? (I believe that a counter-example plus the pigeon-hole principle would yield a periodic counter-example.)