For sure, @stargazer07817 … as implied here and here, I’ve looked at linear combinations of multiple cycle members’ betas, where the number of members included 1, 2, 3, 4 and l itself.
While I’ve poked around a fair amount in this area, probably I’ve been digging in the hedges and missed the spaceship buried in the backyard.
With the idea you mention, you can certainly do better at small l, bringing more vectors under the dashed \delta line. But I haven’t found any pattern that generalizes.
For example, you might guess every cycle has some “signed sum of odd members’ betas” (linear combination with coefficients restricted to 1 or -1) of the form 2^i 3^j y, where always y < \delta(v). This works great for small l, but you eventually reach counter-examples like
v = 11011100001011001100
\ \ \ \ \ \ \ 11101111110110101111
\ \ \ \ \ \ \ 111001110110110001011
\ \ \ \ \ \ \ 111010
The simple difference-method, as it exists, is at least understood in terms of the shared prefix between w and u, so we know something about how and where it works at large l. Maybe it would help to think logically about what happens when more than two vectors’ betas get combined.
Also, counter-examples to any proposal (like the one above) might appear, but then might eventually disappear, at higher l, overwhelmed by the rocketing 2^l - 3^s. We currently have almost no grasp on the asymptotics.