I was at Wayne State this weekend. I agreed to fill in for someone at the last minute, and I used these slides. They might not be totally self-explanatory, but you can look at them if you’d like.

It also contains a recent discovery that the exponent of 1/8 is mistaken. I found a mistake in the computation of the exponent. Before, it was thought that it only depended on 3 or 4 things, but now it seems that it depends on 8 or 12 things running through the whole paper in technical places and depending on several different constants. I ran the numbers a few times and didn’t get the same exponent twice, but I was getting things between 1/88 and 1/92; let’s call it 1/100. There will be a new version, either in the arXiv or submitted to a journal directly which addresses this update.


So there are some small errors in v.1 of the recent paper. I will probably find even more later. They are typographical rather than substantial, and maybe one or two places could use a few more words. Perhaps these typographical problems are of no consequence to an author trying to read it, but it may be a real hurdle for an uninitiated reader trying to make sense of it line by line. Hopefully we’ll fix a few things soon and either release it v2 or put it all in another version we’ll try to get published. If something in v1 looks like a misprint but you’re not sure, let me know.

Our new, improved paper is online now. Not only is the estimate much sharper thanks to one nice little lemma using Hardy space theory and another using something we call “analytic tiling”*, but you may even be able to read this one.

A 3^{-n}-neighborhood of a 1-dimensional Sierpinski gasket decays in Buffon needle probability at least as fast as C/{n^p}.

* – “Analytic tiling” is essentially our fancy name for the observation that if three unit complex numbers sum to 0, they form an equilateral triangle, and therefore, their third powers sum to a number of size 3. The name is motivated by the role tiling played in Laba and Zhai’s paper on product Cantor sets.