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sniffnoyI posted a series of tweets about this the other day, but, y'know what, let's write this up a little more properly on DW.
Whenever I see a theorem or conjecture about a gap, between the smallest element of a set and the second-smallest, I ask if this is the start of a well-ordering. Obviously, this applies in reverse too; if there's a gap between the largest and second-largest, I ask if it's a reverse well-ordering.
(And remember, when asking about your set's order type, don't forget to ask if it's closed, or to consider the closure if it's not!)
What are some examples of this? Well, of course, there's all my theorems and/or conjectures on integer complexity and addition chains (and all sorts of variants), but I'll skip going into that for now, because I've discussed that at length here already.
There's the group theory problem I mentioned in my last entry. That's certainly a good one; the numerical data looks promising there.
Here are some others I can think of, off the top of my head:
The 1/3-2/3 conjecture: Given a finite poset P, define δ(P) to be the maximum over all pairs of distinct elements (x,y) of the minimum of the fraction of linearizations in which x<y and the fraction of linearizations in which y<x. (So, δ(P)=0 iff P is already a total order.) Then the 1/3-2/3 conjecture says that any P with δ(P)>0 in fact has δ(P)≥1/3. I.e., it's a gap conjecture. Could the values of δ be well-ordered?
(Maybe they can't! I don't know a lot about this sort of thing, maybe there's already a counterexample.)
(EDIT October 2021: Yup, they're not.)
What about various number-theoretic sets of real algebraic integers? Like, there's Lehmer's conjecture, which conjectures that there's a gap between 1 and the next-smallest Mahler measure. However, Mahler measures can't be well-ordered, because the include the Pisot numbers. That said, we can still ask about order type! In particular, the order type of the Pisot numbers is known, and it's a nifty one! Though it begins with an initial ω, it isn't a well-order after that. Still, like I said, it's a nifty order.
In fact, I'd conjecture that if you considered integer complexity or addition chains with subtraction allowed, and considered the defects arising from those, and then either took the closure or added whole numbers (because those should probably give the same set), that the result would be order-isomorphic to the Pisot numbers. But I'm not trying to prove that right now; I tried briefly and found that, total orders get much harder once they're not well-orders! Someday, I hope to get back to this, but certainly not now.
Anyway, they won't be well-ordered, but one could still ask about order types of (or closures of) Salem numbers or Mahler measures or Perron numbers.
(EDIT October 2021: No, not really; it turns out that each Pisot number is approached on boy sides by Salem numbers, and so Salem numbers are not well-ordered either; and Mahler measures and Perron numbers include the Salem and Pisot numbers and so are definitely not well-ordered either. Indeed, I already noted this for Mahler measures above...? This entry seems a bit inconsistent...)
Of course maybe you get nothing like that. Like in the theorem mentioned in this MathOverflow answer, where you start out with a gap, then another gap, but then you just get all real numbers past that point. What can I say, nothing's guaranteed.
Still, I think this is a question people should ask more often. So: What are other examples of sets of numbers with gap theorems, that we could ask if they're well-ordered? I'm sure I must have missed a ton.
-Harry
Whenever I see a theorem or conjecture about a gap, between the smallest element of a set and the second-smallest, I ask if this is the start of a well-ordering. Obviously, this applies in reverse too; if there's a gap between the largest and second-largest, I ask if it's a reverse well-ordering.
(And remember, when asking about your set's order type, don't forget to ask if it's closed, or to consider the closure if it's not!)
What are some examples of this? Well, of course, there's all my theorems and/or conjectures on integer complexity and addition chains (and all sorts of variants), but I'll skip going into that for now, because I've discussed that at length here already.
There's the group theory problem I mentioned in my last entry. That's certainly a good one; the numerical data looks promising there.
Here are some others I can think of, off the top of my head:
The 1/3-2/3 conjecture: Given a finite poset P, define δ(P) to be the maximum over all pairs of distinct elements (x,y) of the minimum of the fraction of linearizations in which x<y and the fraction of linearizations in which y<x. (So, δ(P)=0 iff P is already a total order.) Then the 1/3-2/3 conjecture says that any P with δ(P)>0 in fact has δ(P)≥1/3. I.e., it's a gap conjecture. Could the values of δ be well-ordered?
(Maybe they can't! I don't know a lot about this sort of thing, maybe there's already a counterexample.)
(EDIT October 2021: Yup, they're not.)
What about various number-theoretic sets of real algebraic integers? Like, there's Lehmer's conjecture, which conjectures that there's a gap between 1 and the next-smallest Mahler measure. However, Mahler measures can't be well-ordered, because the include the Pisot numbers. That said, we can still ask about order type! In particular, the order type of the Pisot numbers is known, and it's a nifty one! Though it begins with an initial ω, it isn't a well-order after that. Still, like I said, it's a nifty order.
In fact, I'd conjecture that if you considered integer complexity or addition chains with subtraction allowed, and considered the defects arising from those, and then either took the closure or added whole numbers (because those should probably give the same set), that the result would be order-isomorphic to the Pisot numbers. But I'm not trying to prove that right now; I tried briefly and found that, total orders get much harder once they're not well-orders! Someday, I hope to get back to this, but certainly not now.
Anyway, they won't be well-ordered, but one could still ask about order types of (or closures of) Salem numbers or Mahler measures or Perron numbers.
(EDIT October 2021: No, not really; it turns out that each Pisot number is approached on boy sides by Salem numbers, and so Salem numbers are not well-ordered either; and Mahler measures and Perron numbers include the Salem and Pisot numbers and so are definitely not well-ordered either. Indeed, I already noted this for Mahler measures above...? This entry seems a bit inconsistent...)
Of course maybe you get nothing like that. Like in the theorem mentioned in this MathOverflow answer, where you start out with a gap, then another gap, but then you just get all real numbers past that point. What can I say, nothing's guaranteed.
Still, I think this is a question people should ask more often. So: What are other examples of sets of numbers with gap theorems, that we could ask if they're well-ordered? I'm sure I must have missed a ton.
-Harry
