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

We study statistics of tiles in random incommensurable Kakutani sequences of partitions in $\mathbb{R}^d$. We provide explicit formulas that illustrate the dependence on the combinatorial structure, the volumes of the participating tiles and the entropy of the partitions in the underlying random substitution system. These improve previous results for non-random Kakutani partitions and multiscale substitution tilings, and imply a gap distribution formula for Delone sets associated with multiscale substitution tilings of the real line.Comment: 13 pages, 3 figures. The abstract has been changed. To appear in Journal of Mathematical Analysis and Application

Sum of Two Squares - Pair Correlation and Distribution in Short Intervals

In this work we show that based on a conjecture for the pair correlation of integers representable as sums of two squares, which was first suggested by Connors and Keating and reformulated here, the second moment of the distribution of the number of representable integers in short intervals is consistent with a Poissonian distribution, where "short" means of length comparable to the mean spacing between sums of two squares. In addition we present a method for producing such conjectures through calculations in prime power residue rings and describe how these conjectures, as well as the above stated result, may by generalized to other binary quadratic forms. While producing these pair correlation conjectures we arrive at a surprising result regarding Mertens' formula for primes in arithmetic progressions, and in order to test the validity of the conjectures, we present numericalz computations which support our approach.Comment: 3 figure

Bounded displacement non-equivalence in substitution tilings

Frettlöh D, Smilansky Y, Solomon Y. Bounded displacement non-equivalence in substitution tilings. Journal of Combinatorial Theory, Series A. 2021;177: 105326.In the study of aperiodic order and mathematical models of quasicrystals, questions regarding equivalence relations on Delone sets naturally arise. This work is dedicated to the bounded displacement (BD) equivalence relation, and especially to results concerning instances of non-equivalence. We present a general condition for two Delone sets to be BD non-equivalent, and apply our result to Delone sets associated with tilings of Euclidean space. First we consider substitution tilings, and exhibit a substitution matrix associated with two distinct substitution rules. The first rule generates only periodic tilings, while the second generates tilings for which any associated Delone set is non-equivalent to any lattice in space. As an extension of this result, we introduce arbitrarily many distinct substitution rules associated with a single matrix, with the property that Delone sets generated by distinct rules are non-equivalent. We then turn to the study of mixed substitution tilings, and present a mixed substitution system that generates representatives of continuously many distinct BD equivalence classes. (C) 2020 Elsevier Inc. All rights reserved