Uniform distribution of Kakutani partitions generated by substitution schemes

Statistics and gap distributions in random Kakutani partitions and multiscale substitution tilings

“…The standard construction of substitution tilings, which includes the well-known Penrose and pinwheel tilings, is related to such sequences, see [BG] for a comprehensive discussion. In the case of primitive (not necessarily normalized) fixed scale schemes, in which all the participating scales in (2.1) are identical, it is well-known that the Perron-Frobenius theorem implies the following formulas for the tile frequencies, as explained also in [Sm,Theorem 6.12]. These may be compared with the analogous formulas for incommensurable Kakutani sequences of partitions in Corollary 2.4.…”

“…As noted above, general formulas for tile frequencies in the non-random case appeared in [SS,§7], see also [Sm,Theorem 1.13], using a similar strategy to the one applied in the current contribution. While allowing for computations in specific cases and examples, the formulas remained somewhat opaque, and were not given explicitly in terms of the associated matrices or in terms of density functions, making them challenging to decipher and giving little additional insight to the way the frequencies depend on the generating substitution scheme.…”

Discrepancy and rectifiability of almost linearly repetitive Delone sets

“…A patch of an irreducible incommensurable multiscale substitution tiling in R 2 is illustrated below in Figure 1. For more details, examples, illustrations and equivalent definitions of incommensurability, multiscale substitutions schemes and the geometric objects they generate, the reader is referred to [Sm1] and [SS1].…”

An infinite interval version of the $α$-Kakutani equidistribution problem

“…Our interest in this problem, and the starting point for our analysis, began with the very elegant work of Smilansky [17].…”

“…To make use of this renewal equation, just as in [2,8,17] it is necessary to consider two cases which behave somewhat differently. These cases correspond to, for example, the α-Kakutani schemes for α = 1/3 and α = 1/2, as described in the introduction.…”