Normalise $\|F\|_{L^d({\bf R}^n)} = 1$. The above arguments will take care of the range when (say), since the number of such $k$ is only logarithmic in and one can move from to with a loss that can be absorbed in the factor. The remaining extreme ranges or can then be handled by trivial estimates (the former by bounding by ; the latter by bounding by ).

]]>Can you give more details on the procedure from estimate (2) to

It seems to me that just breaking up into level sets via dyadic decomposition will not work. In fact, we have

where the estimate (2) has been used. Our target is to bound the above quantity by

Now, we need Hölder’s inequality from to , which is not aviable.

Thanks for the reply, I didn’t notice the difficulty with the exponent of d at first glance. Speaking of exponents, I think there should be a in the statement of Conjecture 3, as well as in the first line of the proof of Proposition 7.

Thanks again for providing such clear concise expositions regarding Kakeya, they’re truly wonderful to read.

]]>Thanks for the corrections! The random rotations trick is used where it is because of the placement of the exponents. If one replaced the factor in Conjecture 2 by the weaker factor , then (2) and Conjecture 2 would be equivalent without the random rotations trick, but then one would need that trick to deduce Conjecture 3 from Conjecture 2.

]]>]]>

Thanks for the wonderful post. A few possible corrections:

I’m a bit confused with the change from to going from (2) to Conjecture 2. Also, might be enlarged to become .

Also, shouldn't the random rotation trick be used to show Conjecture 2 Conjecture 3? I don’t see why it is needed to prove (2) => Conjecture 2, can’t that be proved directly?

Thanks again.

]]>To deduce (2) from conjecture 2, first observe that the claim is easy if is very big or very small (e.g. larger than or less than for some large C depending only on n) because one can use trivial estimates such as and in those cases.

For similar reasons, one can also throw away those tubes for which is very small (less than for some large C). Once one does all this, there are now only a logarithmic number of ranges that the could fall into, where is a dyadic number. For each such range, one applies Conjecture 2, which will establish (2) for that particular subcollection of tubes after some algebra (at one point one has to use the fact that ); then one sums in the dyadic ranges, absorbing the logarithmic loss into the factor on the right-hand side.

To see the translation and rotation symmetry of the Kakeya problem in this setting, one has to translate and rotate both E and the tubes .

Thanks for the correction!

]]>Two questions and one corrections.

1. Can you say a little more on the derivation of the estimate (2) from Conjecture 2?

2. If formulating this conjecture in terms of

for the maximal function

.

Then the translation and rotation symmetries are quickly seen. How to reflect these informaton in the formulatons of Conjecture 1, ect? Does it suffice by saying that: fixing a set E; then for any collection of tubes, …?

(3) In the last 8 lines in the argument of Proposition 5, and

may need to be exchanged.