I already discussed the very interesting mathematics related to the Bohr-Pál Theorem that I learnt about while writing my preprint with W. Sawin on the support of Kloosterman path. This also led us (very) tangentially to what turns out to be an open geometric problem: Mihalik and Wieczorek asked whether there exists a continuous function which sends every interval in to a convex subset of , and whose image is not contained in an affine line in . In other words: suppose is continuous and satisfies the intermediate value property (in the sense that for any , any point on the segment between and is of the form for some between and ); is the image of contained in some affine line?
The existence of such functions may seem unlikely, but experience shows that being unlikely to exist has rarely stopped functions from actually existing.
The arithmetic relevance of a hypothetical Mihalik-Wieczorek function is that, by adapting Sahakyan’s argument (as discussed before), it seems that we would be able to construct a function in the support of Kloosterman paths that has image containing an open set, answering one of our lingering questions.
However, the question remains apparently open. The best known result (that we know about) is due to Pach and Rogers (1982), and independently Vince and Wilson (1984): there exists , with image not contained in an affine line, such that and are convex for all . (Note that Vince and Wilson conjecture at the end of their paper that a Mihalik-Wieczorek curve does not exist; however their argument is based on a weaker conjecture that this existence would imply, and that statement in turn is actually true, as was explicitly stated by Pach and Rogers at the end of their article; see this American Math. Monthly problem.).
Update (May 13, 2018) I’ve written a short note that checks in detail the statement above: if a Mihalik-Wieczorek curve exists, then there exists a space-filling curve in the support of the Kloosterman paths. (More precisely, this is done without the symmetry requirement of the Kloosterman paths, but these can be ensured by scaling the given curve to and then reflecting it on ). This note can also be seen as a detailed account of Sahakian’s Theorem…