Paper update

Just before the holidays, Malabika Pramanik and I finished our paper Arithmetic progressions in sets of fractional dimension and posted it on the arXive. Or at least we thought we’d finished it. As it turns out, there will have to be several corrections before we submit it for publication. Be warned that this won’t make much sense to you if you haven’t at least browsed the paper.

• Theorem 1.2 (i.e. there is a subset of R of Hausdorff dimension 1 which contains no 3-term arithmetic progressions) turns out not to be new. A stronger result had been proved by Tamas Keleti back in 1998. Thanks to Mihalis Kolountzakis for pointing it out!
• We have found a mistake in the proof of our Theorem 1.4 (the modified version of our main result, Theorem 1.3, with assumptions weakened to accommodate Brownian motion examples). We’re working on it but we don’t know yet whether it can be fixed. This applies to Theorem 1.4 only – Theorem 1.3 itself is just fine.
• On the other hand, it looks like our results do apply to Salem’s examples after all. We’re double-checking this.

We are working on the corrections and we will try to upload the revised version by early next week.

Filed under mathematics: research

2 responses to “Paper update”

1. Dear Prof. Laba,

I find interest in your paper with M. Pramanik, and I really hope that the problems could be fixed.

I would like to comment, that the remark after the formulation of Theorem 1.3 (on page 2) is perhaps not completely accurate. Namely, I am not sure that if (B) holds with beta=1, then the measure must be absolutely continuous. This is because it seems to contradict a well-known theorem of Ivashev-Musatov.

I look forward to meeting you in the Fields Institute thematic program.

Best regards,
Nir Lev.

2. Dear Nir,

Thanks for the comment! You are right – we were sloppy with the remark after Theorem 1.3. What we really use in our proof is that alpha is strictly less than one. If alpha is 1, the proof does not go through (for pretty obvious reasons), but in this case the measure is absolutely continuous. The proof does go through if beta =1, so this case does not bother us. Thanks for pointing out the interesting fact. We have already revised the paper accordingly.

It looks like we were indeed able to fix the problem. The incorrect part was the “restriction via subharmonic interpolation” result, which we applied to the Brownian motion examples. We don’t know how to fix that, but we’ve found an alternative approach to that class of examples that seems to work better. We may be able to upload the revised version tomorrow if all goes well [knock on wood].

Best, Izabella