Malabika Pramanik and I have just finished revising our paper on arithmetic progressions in sets of fractional dimension. The new version is now available on my web page. We have also uploaded it to the arXiv.
In the new version, we have deleted our “iterated Behrend” construction and refer to Keleti’s result instead. We have also revised substantially the section on Brownian images. Instead of the “restriction via subharmonic interpolation” result, which was incorrect and has been deleted, we now prove directly a lower bound on the $\Lambda$ quantity and deduce the result about arithmetic progressions.
(Yes, I know that one can use LATEX in WordPress. I’ll figure out how to do it some other time. Right now I’m done with work for today.)