three gap theorem
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2020 ◽  
Vol 54 (1) ◽  
pp. 31-37
Author(s):  
Christian Weiss

The Three Gap Theorem states that there are at most three distinct lengths of gaps if one places n points on a circle, at angles of z, 2z, … nz from the starting point. The theorem was first proven in 1958 by Sós and many proofs have been found since then. In this note we show how the Three Gap Theorem can easily be deduced by using Rauzy-Veech induction.


2017 ◽  
Vol 124 (8) ◽  
pp. 741 ◽  
Author(s):  
Jens Marklof ◽  
Andreas Strömbergsson

2016 ◽  
Vol 56 (6) ◽  
pp. 462-471 ◽  
Author(s):  
Zuzana Masáková ◽  
Edita Pelantová ◽  
Štěpán Starosta

We focus on a generalization of the three gap theorem well known in the framework of exchange of two intervals. For the case of three intervals, our main result provides an analogue of this result implying that there are at most 5 gaps. To derive this result, we give a detailed description of the return times to a subinterval and the corresponding itineraries.


2016 ◽  
Vol 12 (07) ◽  
pp. 1743-1764 ◽  
Author(s):  
Geremías Polanco ◽  
Daniel Schultz ◽  
Alexandru Zaharescu

The well-known Three Gap Theorem states that there are at most three gap sizes in the sequence of fractional parts [Formula: see text]. It is known that if one averages over [Formula: see text], the distribution becomes continuous. We present an alternative approach, which establishes this averaged result and also provides good bounds for the error terms.


2008 ◽  
Vol 136 (1) ◽  
pp. 175-190
Author(s):  
Ian Biringer ◽  
Benjamin Schmidt

2008 ◽  
Vol 08 (01) ◽  
pp. 115-125 ◽  
Author(s):  
G. NICOLIS ◽  
V. BALAKRISHNAN ◽  
C. NICOLIS

We consider the distribution of the maximum for finite, deterministic, periodic and quasiperiodic sequences, and contrast the extreme value distributions in these cases with the classical results for iidrv's. A significant feature in the case of deterministic sequences is a multi-step structure for the distribution function. The extreme value distribution for the circle map with an irrational parameter is obtained in closed form with the help of the three-gap theorem for the map Xj+1 = (Xj + a) mod 1 where a ∈ (0,1) is an irrational number.


Author(s):  
Tony van Ravenstein

AbstractThis paper is concerned with the distribution of N points placed consecutively around the circle by an angle of α. We offer a new proof of the Steinhaus Conjecture which states that, for all irrational α and all N, the points partition the circle into arcs or gaps of at least two, and at most three, different lengths. We then investigate the partitioning of a gap as more points are included on the circle. The analysis leads to an interesting geometrical interpretation of the simple continued fraction expansion of α.


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