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Let X^{(k)}(t) = (X_1(t), ..., X_k(t)) denote a k-vector of i.i.d. random variables, each taking the values 1 or 0 with respective probabilities p and 1-p. As a process indexed by non-negative t, $X^{(k)}(t)$ is constructed--following Benjamini, Haggstrom, Peres, and Steif (2003)--so that it is strong Markov with invariant measure ((1-p)delta_0+pdelta_1)^k. We derive sharp estimates for the probability that ``X_1(t)+...+X_k(t)=k-ell for some t in F, where F subset [0,1] is nonrandom and compact. We do this in two very different settings: (i) Where ell is a constant; and (ii) Where ell=k/2, k is even, and p=q=1/2. We prove that the probability is described by the Kolmogorov capacitance of F for case (i) and Howroyds 1/2-dimensional box-dimension profiles for case (ii). We also present sample-path consequences, and a connection to capacities that answers a question of Benjamini et. al. (2003)
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Every partial function from bit strings of a given length to bit strings of a possibly different given length can be computed by a finite instruction sequence that contains only instructions to set and get the content of Boolean registers, forward jump instructions, and a termination instruction. We look for an equivalence relation on instruction sequences of this kind that captures to a reasonable degree the intuitive notion that two instruction sequences express the same algorithm.
In this paper we show how the cross-disciplinary transfer of techniques from Dynamical Systems Theory to Number Theory can be a fruitful avenue for research. We illustrate this idea by exploring from a nonlinear and symbolic dynamics viewpoint certain patterns emerging in some residue sequences generated from the prime number sequence. We show that the sequence formed by the residues of the primes modulo $k$ are maximally chaotic and, while lacking forbidden patterns, display a non-trivial spectrum of Renyi entropies which suggest that every block of size $m>1$, while admissible, occurs with different probability. This non-uniform distribution of blocks for $m>1$ contrasts Dirichlets theorem that guarantees equiprobability for $m=1$. We then explore in a similar fashion the sequence of prime gap residues. This sequence is again chaotic (positivity of Kolmogorov-Sinai entropy), however chaos is weaker as we find forbidden patterns for every block of size $m>1$. We relate the onset of these forbidden patterns with the divisibility properties of integers, and estimate the densities of gap block residues via Hardy-Littlewood $k$-tuple conjecture. We use this estimation to argue that the amount of admissible blocks is non-uniformly distributed, what supports the fact that the spectrum of Renyi entropies is again non-trivial in this case. We complete our analysis by applying the Chaos Game to these symbolic sequences, and comparing the IFS attractors found for the experimental sequences with appropriate null models.
We investigate the problem of embedding infinite binary sequences into Bernoulli site percolation on $mathbb{Z}^d$ with parameter $p$, known also as percolation of words. In 1995, I. Benjamini and H. Kesten proved that, for $d geq 10$ and $p=1/2$, all sequences can be embedded, almost surely. They conjectured that the same should hold for $d geq 3$. In this paper we consider $d geq 3$ and $p in (p_c(d), 1-p_c(d))$, where $p_c(d)<1/2$ is the critical threshold for site percolation on $mathbb{Z}^d$. We show that there exists an integer $M = M (p)$, such that, a.s., every binary sequence, for which every run of consecutive {0s} or {1s} contains at least $M$ digits, can be embedded.
We study two types of dynamical extensions of Lucas sequences and give elliptic solutions for them. The first type concerns a level-dependent (or discrete time-dependent) version involving commuting variables. We show that a nice solution for this system is given by elliptic numbers. The second type involves a non-commutative version of Lucas sequences which defines the non-commutative (or abstract) Fibonacci polynomials introduced by Johann Cigler. If the non-commuting variables are specialized to be elliptic-commuting variables the abstract Fibonacci polynomials become non-commutative elliptic Fibonacci polynomials. Some properties we derive for these include their explicit expansion in terms of normalized monomials and a non-commutative elliptic Euler--Cassini identity.
An infinite sequence of real random variables $(xi_1, xi_2, dots)$ is said to be rotatable if every finite subsequence $(xi_1, dots, xi_n)$ has a spherically symmetric distribution. A celebrated theorem of Freedman states that $(xi_1, xi_2, dots)$ is rotatable if and only if $xi_j = tau eta_j$ for all $j$, where $(eta_1, eta_2, dots)$ is a sequence of independent standard Gaussian random variables and $tau$ is an independent nonnegative random variable. Freedmans theorem is equivalent to a classical result of Schoenberg which says that a continuous function $phi : mathbb{R}_+ to mathbb{C}$ with $phi(0) = 1$ is completely monotone if and only if $phi_n: mathbb{R}^n to mathbb{R}$ given by $phi_n(x_1, ldots, x_n) = phi(x_1^2 + cdots + x_n^2)$ is nonnegative definite for all $n in mathbb{N}$. We establish the analogue of Freedmans theorem for sequences of random variables taking values in local fields using probabilistic methods and then use it to establish a local field analogue of Schoenbergs result. Along the way, we obtain a local field counterpart of an observation variously attributed to Maxwell, Poincare, and Borel which says that if $(zeta_1, ldots, zeta_n)$ is uniformly distributed on the sphere of radius $sqrt{n}$ in $mathbb{R}^n$, then, for fixed $k in mathbb{N}$, the distribution of $(zeta_1, ldots, zeta_k)$ converges to that of a vector of $k$ independent standard Gaussian random variables as $n to infty$.
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