Buffon’s needles: a very short expository note

One of the things I have been working on in the last few years is the Favard length problem. The question is to estimate the average length of a 1-dimensional projection of (a finite iteration of) a 1-dimensional self-similar Cantor set in the plane. My work with Zhai, and especially with Bond and Volberg, has pointed to connections with classical questions in number theory, including tilings of the integers, diophantine approximation of logarithms of algebraic numbers, and vanishing sums of roots of unity.

If you would like to find out a little bit more about this, but don’t necessarily feel like reading long technical papers that rely on several other long technical papers, then this very short expository note (3 pages plus short bibliography) might be for you. It was written for the CMS Notes and has about the length they require. I found myself wanting to make it longer, if only to include more references to the history and context of the problem. (I never even mentioned Comte de Buffon.) I might write a more substantial expository paper when I have the time.

Author: Izabella Laba

Mathematics professor at UBC. My opinions are, obviously, my own.

2 thoughts on “Buffon’s needles: a very short expository note”

  1. In Theorem 1.1, did you mean p<1/6? In any case you certainly need p<1.

    I'd be interested to see a more substantial exposition, in particular one that expanded on why "We would like to understand the 1-dimensional projections of K."

  2. Yes, that should be p < 1/6. And yes, the context is one of the things that I would have liked to elaborate on.

    Edited to add a little bit more about the context: on the one hand, there is the connection to analytic capacity. I’m not going to try to write an exposition of this here on the spot, but Xavier Tolsa mentions it in his ICM 2006 Proceedings paper. On the other hand, there are many related questions in ergodic theory. For example, there is a long-standing conjecture of Furstenberg that any projection of the 1-dimensional “Sierpinski gasket” in a direction with irrational slope has Hausdorff dimension 1. (This is still open, with the best known results due to Swiatek and Veerman. The “tiling” connection was actually first explored in a paper by Kenyon motivated by this conjecture.)

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