Take a point on the unit circle and rotate it N times by a fixed angle. The N points thus generated partition the circle into N intervals. A beautiful fact, first conjectured by Hugo Steinhaus in the 1950s and proved independently by Vera Sós, János Surányi and Stanisław Świerczkowski, is that for any choice of N, no matter how large, these intervals can have at most three distinct lengths. In this lecture I will explore an interpretation of the three gap theorem in terms of the space of Euclidean lattices, which will produce various new results in higher dimensions, including gaps in the fractional parts of linear forms and nearest neighbour distances in multi-dimensional Kronecker sequences. The lecture is based on joint work with Alan Haynes (Houston) and Andreas Strömbergsson (Uppsala).
1. Wikipedia, https://en.wikipedia.org/wiki/Three-gap_theorem
2. J. Marklof and A. Strömbergsson, The three gap theorem and the space of lattices, American Mathematical Monthly 124 (2017) 741-745 https://people.maths.bris.ac.uk/~majm/bib/threegap.pdf
3. A. Haynes and J. Marklof, Higher dimensional Steinhaus and Slater problems via homogeneous dynamics, Annales scientifiques de l'Ecole normale superieure 53 (2020) 537-557 https://people.maths.bris.ac.uk/~majm/bib/steinhaus.pdf
4. A. Haynes and J. Marklof, A five distance theorem for Kronecker sequences, preprint arXiv:2009.08444 https://people.maths.bris.ac.uk/~majm/bib/steinhaus2.pdf
This talk is part of the online Number Theory Web Seminar, and will be streamed live on Zoom.
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Mike Bennett (University of British Columbia)
Philipp Habegger (University of Basel)
Alina Ostafe (UNSW Sydney)