Uniform distribution of Kakutani partitions generated by substitution schemes
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UNIFORM DISTRIBUTION OF KAKUTANI PARTITIONS GENERATED BY SUBSTITUTION SCHEMES BY
Yotam Smilansky∗ Raymond and Beverly Sackler School of Mathematical Sciences Tel Aviv University, Tel Aviv 69978, Israel e-mail: [email protected]
ABSTRACT
Substitution schemes provide a classical method for constructing tilings of Euclidean space. Allowing multiple scales in the scheme, we introduce a rich family of sequences of tile partitions generated by the substitution rule, which include the sequence of partitions of the unit interval considered by Kakutani as a special case. Using our recent path counting results for directed weighted graphs, we show that such sequences of partitions are uniformly distributed, thus extending Kakutani’s original result. Furthermore, we describe certain limiting frequencies associated with sequences of partitions, which relate to the distribution of tiles of a given type and the volume they occupy.
1. Introduction and main results 1.1. The Kakutani splitting procedure. The splitting procedure introduced by Kakutani in [Ka] and the associated sequences of partitions {πm } of the unit interval I = [0, 1] are defined as follows: Fix a constant α ∈ (0, 1) and define the first element of the sequence to be the trivial partition π0 = I. Next, define π1 to be the partition of I into two subintervals [0, α] and [α, 1], that is, into two intervals of disjoint interior, one of length α and one of length 1 − α. Assuming πm−1 is defined, define the partition πm by splitting every interval of ∗ Current address: Department of Mathematics, Rutgers University, 110 Frel-
inghuysen Road, Piscataway, NJ 08854, USA. Received June 11, 2018 and in revised form September 22, 2019
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maximal length in πm−1 into two subintervals with disjoint interiors, proportional to the two subintervals that constitute π1 . This sequence is known as the α-Kakutani sequence of partitions, and the procedure that generates it is called the Kakutani splitting procedure. For example, the trivial partition π0 and the first four non-trivial partitions π1 , π2 , π3 and π4 of the 13 -Kakutani sequence of partitions of the unit interval I are illustrated in Figure 1.1.
Figure 1.1. From left to right, the first few partitions in the 13 -Kakutani sequence of partitions of the unit interval. A sequence {πm } of partitions of I is said to be uniformly distributed if for any continuous function f on I k(m) 1 f (tm ) = f (t)dt, i m→∞ k(m) I i=1 lim
where k(m) is the number of intervals in the partition πm and tm i is the right endpoint of the interval i in the partition πm . We remark that the choice of sampling points as the right endpoints of the intervals is arbitrary. Theorem (Kakutani): For any α ∈ (0, 1), the α-Kakutani sequence of partitions is uniformly distributed. Our main result is a proof of uniform distribution for sequences of partitions generated by multiscale substitution schemes in Rd , constituting a generalization of the α-Kakutani sequences considered above. 1.2. Tiles, partitions and m
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