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Apr
29
comment Choice vs. countable choice
I'm in the same minority. I don't find it "obviously true" in any useful sense that the empty set exists, for instance. I just don't know what "exists" means over and above our use of the axiom in mathematics.
Apr
29
comment A question of Erdős on equidistribution
No idea what the answer is, but thanks for drawing my attention to this nice problem. (I'm hoping someone will say that it's still open.) How easy is the counterexample to Khintchine's conjecture?
Apr
28
comment Wanted: A constructive version of a theorem of Furstenberg and Weiss
OK that gives me a better idea of what you are asking -- enough to see that I don't know the answer I'm afraid. (My gut reaction is that what you're asking for looks hard, but I don't know much about this area so this opinion should not be taken seriously.)
Apr
28
comment Wanted: A constructive version of a theorem of Furstenberg and Weiss
It's certainly possible to produce a constructive upper bound. Are you really looking for the best possible upper bound?
Apr
27
comment How to know if somebody else is also working on your problem?
"If the other guy has already completed a certain amount of (say, not yet published) work on that specific topic, knowing this would help you to avoid waisting time to try to re-do something that has already been done (at least with the same methods)." I wouldn't underestimate the value of that kind of waste of time ...
Apr
22
comment A Simple Generalization of the Littlewood Conjecture
I can't quite follow what your argument is, but I would point out that it is a known fact that if a sequence $d_1,d_2,...$ grows geometrically (in the strong sense that there is a positive constant c such that all ratios are at least 1+c) then there must exist a real number s and a positive epsilon such that for every n the distance from $sd_n$ to the nearest integer is at least epsilon.
Apr
22
comment Can curves induced by analytic maps wiggle infinitely across a line?
Very nice. My mistake was to try to use the no-isolated-zeros theorem rather than imitating its proof.
Apr
22
comment Can curves induced by analytic maps wiggle infinitely across a line?
My comment does not imply that the question isn't straightforward. I am struggling to understand your argument, however. Are your "real analytic" functions complex-valued? I should add that the fact that I am struggling to understand your argument is compatible with its being clear and correct.
Apr
22
comment A Simple Generalization of the Littlewood Conjecture
Hmm, I have now thought a bit more and I think this paper is irrelevant -- though it is interesting in its own right. I like your question too.
Apr
22
comment Can curves induced by analytic maps wiggle infinitely across a line?
Oh dear, this question looks as though it has the potential to obsess. I thought I had an answer but it collapsed as I tried to write it down.
Apr
19
comment Subset of the plane that intersects every line exactly twice
If my memory serves, it's open even for G_delta sets (but known that they cannot be F_sigma sets).
Apr
16
comment Pairwise intersecting sets of fixed size
It was meant to be the second or your possibilities, so 2^{k+1} vertices in all.
Apr
10
comment Most 'unintuitive' application of the Axiom of Choice?
Another point I wanted to make was that if your set is chosen using AC, then it's not really clear what it means to choose a random real and ask whether it lands in the set. After all, you haven't said what the set is. So I'm not sure we would get this oscillatory behaviour because I'm not sure it's possible to make mathematical sense of the experiment in the first place. I do like it though ...
Apr
10
comment Most 'unintuitive' application of the Axiom of Choice?
Suppose you construct a non-measurable set of 01-sequences as follows. Call two sequences equivalent if they differ in finitely many places. Choose a sequence from each equivalence class, and then let your set be the set of all sequences that have even symmetric difference with the chosen representative of its equivalence class. It feels as though a random sequence should have probability 1/2 of belonging to the set. And if you change the definition to "symmetric difference has size congruent to 0 mod 100, it feels as though the probability should be 1/100. I'm confused.
Apr
10
comment Most 'unintuitive' application of the Axiom of Choice?
The existence of undetermined games is an easy application of the axiom of choice, and somewhat strange. Another is the existence of a set in the plane that intersects every line exactly twice, though it is not known whether choice is needed for that one.
Apr
8
comment What's a natural candidate for an analytic function that interpolates the tower function?
A smoother way of thinking about the same basic idea would be to define the coefficient a_n to be eta(n)^n, where eta is a function that converges to zero very very slowly. Then the given function will tend to infinity very very quickly. It's then just a question of working out the relationship between the two growth rates and making eta(n) as "nice" as possible.
Apr
6
comment A graph on irrationals where p is adjacent to q if p^q or q^p is rational.
Since the degree of each vertex is at most countable, each component is countable, so the number of components is the cardinality of the continuum.
Apr
1
comment Estimate rate of real correct/wrong from 4 answers quiz.
Without some extra information or hypothesis, the evidence is consistent with everyone who answered "Buzz Lightyear" genuinely believing that, and also consistent with everybody making the "Buzz Aldrin" confusion. But it sounds as though you have some prior distribution in mind, which tells you that if someone answers "Buzz Lightyear" then the probability that they think the answer is a character in Toy Story is almost zero. I don't think you can do without that, so if you really want to understand this example you may need to supplement it with another experiment.
Mar
31
comment Is this a well known NP-complete problem?
AH -- that changes things ...
Mar
31
comment Is this a well known NP-complete problem?
That doesn't quite work if you regard non-edges as having weight zero, but even if you do have that convention you can make them into edges with very large weight instead and then the argument works again.