Substitution Delone Sets

This paper addresses the problem of describing aperiodic discrete structures that have a self-similar or self-affine structure. Substitution Delone set families are families of Delone sets (X_1, ..., X_n) in R^d that satisfy an inflation functional equation under the action of an expanding integer matrix in R^d. This paper studies such functional equation in which each X_i is a discrete multiset (a set whose elements are counted with a finite multiplicity). It gives necessary conditions on the coefficients of the functional equation for discrete solutions to exist. It treats the case where the equation has Delone set solutions. Finally, it gives a large set of examples showing limits to the results obtained.Comment: 34 pages, latex file; some results in Sect 5 rearranged and theorems reformulate

Ternary expansions of powers of 2

Paul Erdos asked how frequently the ternary expansion of 2^n omits the digit 2. He conjectured this happens only for finitely many values of n. We generalize this question to consider iterates of two discrete dynamical systems. The first is over the real numbers, and considers the integer part of lambda 2^n for a real input lambda. The second is over the 3-adic integers, and considers the sequence lambda 2^n for a 3-adic integer input lambda. We show that the number of input values that have infinitely many iterates omitting the digit 2 in their ternary expansion is small in a suitable sense. For each nonzero input we give an asymptotic upper bound on the number of the first k iterates that omit the digit 2, as k goes to infinity. We also study auxiliary problems concerning the Hausdorff dimension of intersections of multiplicative translates of 3-adic Cantor sets.Comment: 28 pages latex; v4 major revision, much more detail to proofs, added material on intersections of Cantor set