80 research outputs found
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
Quantum Pascal's triangle and Sierpinski's carpet
In this paper we consider a quantum version of Pascal's triangle. Pascal's triangle is a well-known triangular array of numbers and when these numbers are plotted modulo 2, a fractal known as the Sierpinski triangle appears. We first prove the appearance of more general fractals when Pascal's triangle is considered modulo prime powers. The numbers in Pascal's triangle can be obtained by scaling the probabilities of the simple symmetric random walk on the line. In this paper we consider a quantum version of Pascal's triangle by replacing the random walk by the quantum walk known as the Hadamard walk. We show that when the amplitudes of the Hadamard walk are scaled to become integers and plotted m
Automatic congruences for diagonals of rational functions
In this paper we use the framework of automatic sequences to study combinatorial sequences modulo prime powers. Given a sequence whose generating function is the diagonal of a rational power series, we provide a method, based on work of Denef and Lipshitz, for computing a finite automaton for the sequence modulo pα, for all but finitely many primes p. This method gives completely automatic proofs of known results, establishes a number of new theorems for well-known sequences, and allows us to resolve some conjectures regarding the Apéry numbers. We also give a second method, which applies to an algebraic sequence modulo pα for all primes p, but is significantly slower. Finally, we show that a broad range of multidimensional sequences possess Lucas products modulo p
An isomorphism between the p-adic integers and a ring associated with a tiling of N-space by permutohedra
AbstractThe classical lattice A∗n, whose Voronoi cells tile Euclidean n-space by permutohedra, can be given the generalized balance ternary ring structure GBTn in a natural way as a quotient ring of Z[x]. The ring GBTn can also be considered as the set of all finite sequences s0 s1…sk, with si ∈ GBTn⧸αGBTn for all i, where α is an appropriately chosen element in GBTn. The extended generalized balance ternary (EGBTn) ring consists of all such infinite sequences. A primary goal of this paper is to prove that if 2n+1−1 and n+1 are relatively prime, then EGBTn is isomorphic as a ring to the (2n+1−1)-adic integers
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