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 nearest neighbour distances in multi-dimensional
Kronecker sequences, free flights in the Lorentz gas, and quantum spectra
of harmonic oscillators. The lecture is based on joint work with Alan
Haynes (Houston) and Andreas Strömbergsson (Uppsala).
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Wikipedia, https://en.wikipedia.org/wiki/Three-gap_theorem
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J. Marklof and A. Strömbergsson,
The three gap theorem and the space of lattices,
American Mathematical Monthly 124 (2017) 741-745
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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
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A. Haynes and J. Marklof,
A five distance theorem for Kronecker sequences,
preprint arXiv:2009.08444
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