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.