The idea of the proof was to prove that has certain properties with sufficiently high probability that when we condition on the sum being zero it still has those properties. The main properties are the following two.

1. .

2. The characteristic function of is very small when are not both small.

I think that in the rest of the argument I don’t use probabilistic arguments, but just these properties. So what I have written is not expressed clearly but I think it is correct.

What I think I should have done is got to the end of Section 5 and then fixed an that had the desired properties, together with a sum that is equal to , noting that almost all random -sided dice have this property (which is the main content of Section 5). After that, probabilistic arguments would be banned and the rest of the paper should be deterministic. I need to check, but I think that the only reason certain statements after Section 5 are true only with high probability is that they make use of statements in Section 5 that themselves hold with high probability.

So, to summarize, thank you for picking that up, and at the moment I am fairly confident that it will be easy to correct.

]]>It still needs a bibliography, some comments about the experimental evidence for the stronger quasirandomness conjecture, and I also think we should try to solve two further problems. The first is the question of whether we can prove rigorously that the stronger quasirandomness conjecture is false, and the second is whether we can use similar techniques to prove the weaker conjecture for the multisets model. And one other thing that I very much want to add is a rigorous proof that if you just take random elements of , then which die wins is with very high probability determined by the sum of the faces of the dice: that is, the die with the bigger sum almost certainly wins. I think that should in fact be quite a lot easier than what is proved in the draft so far — it just needs doing, as it explains why in the unconstrained model the experimental evidence suggests that one gets transitivity almost all the time.

]]>“Lake Wobegon Dice,” Jorge Moraleda and David G. Stork, College Mathematics Journal 43(2):152–159, 2012

]]>Maybe for instance by pulling on the thread that the non-uniform weighting changes the weights equally on (die A) and (inverse die A)? ]]>

“Then for around (in the torus topology),

there exists a constant such that

”

That final bound should be , and unfortunately that weakens the bound to the point where it is no longer good enough. However, that just means that looking at successive differences is insufficient, and I’m pretty sure that I am on track to complete the proof with a more complicated argument that looks at longer-range differences. I’m travelling at the moment, so it will probably take me a week or two to have a complete draft of the write-up.

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