The Littlewood conjecrure
I don't believe the Littlewood conjecture will be proved any time soon. Proving that
liminf n||nα|| ||nβ|| = 0
for ALL α and β is beyond our reach. Yes, it's cool that it's been proved that it can only fail for a very small subset of (α, β) but that's just not it.
The most amazing fact for me here is that NOT A SINGLE PAIR OF BADLY APPROXIMABLE IRRATIONAL (α, β) is known for which it holds.
Well, let's take α=(√5+1)/2, the golden ratio, and β=√2+1, say. For the former it makes sense to take n to be the Fibonacci sequence (F_n), as F_{n+1}/F_n well approximates α.
But what about ||F_n β||? Well, here lies the problem. It doesn't appear to vanish. Somehow we need to replace F_n with a sequence that works for both α and β.
How? That's a million dollar question. What do these seemingly different quadratic surds have in common?
liminf n||nα|| ||nβ|| = 0
for ALL α and β is beyond our reach. Yes, it's cool that it's been proved that it can only fail for a very small subset of (α, β) but that's just not it.
The most amazing fact for me here is that NOT A SINGLE PAIR OF BADLY APPROXIMABLE IRRATIONAL (α, β) is known for which it holds.
Well, let's take α=(√5+1)/2, the golden ratio, and β=√2+1, say. For the former it makes sense to take n to be the Fibonacci sequence (F_n), as F_{n+1}/F_n well approximates α.
But what about ||F_n β||? Well, here lies the problem. It doesn't appear to vanish. Somehow we need to replace F_n with a sequence that works for both α and β.
How? That's a million dollar question. What do these seemingly different quadratic surds have in common?