Denote by x a random innnite path in the graph of Pascal's triangle (left and right turns are selected independently with xed probabilities) and by d n (x) the binomial coeecient at the n'th level along the path x. Then for a dense G set of in the unit interval, fd n (x)g is almost surely dense but not uniformly distributed modulo 1.

Brother Alfred has characterized primes dividing every Fibonacci sequence [2] based on their period and congruence class mod 20. More recently, in [4] Ballot and Elia have described the set of primes dividing the Lucas sequence, meaning they divide some term of the sequence. Our purpose here is to extend the results of the former paper utilizing the methods of the latter. In particular we will ...

We study the surface magnetization of aperiodic Ising quantum chains. Using fermion techniques, exact results are obtained in the critical region for quasiperiodic sequences generated through an irrational number as well as for the automatic binary Thue-Morse sequence and its generalizations modulo p. The surface magnetization exponent keeps its Ising value, βs = 1/2, for all the sequences stud...

The discriminator of an integer sequence s = (s(i))i 0, introduced by Arnold, Benkoski, and McCabe in 1985, is the map Ds(n) that sends n 1 to the least positive integer m such that the n numbers s(0), s(1), . . . , s(n 1) are pairwise incongruent modulo m. In this note we consider the discriminators of a certain class of sequences, the k-regular sequences. We compute the discriminators of two ...

Given a morphism h prolongable on a and an integer p, we present an algorithm that calculates which letters occur infinitely often in congruent positions modulo p in the infinite word h(a). As a corollary, we show that it is decidable whether a morphic word is ultimately p-periodic. Moreover, using our algorithm we can find the smallest similarity relation such that the morphic word is ultimate...

A sequence ofreal numbers (xn) is Benford if the significands, i.e., the fraction parts in the floating-point representation of (x ), are distributed logarithmically. Similarly, a discrete-time irreducible and aperiodic fi­ n nite-state Markov chain with transition probability matrix P and limiting matrix P' is Benford if every com­ ponent of both sequences of matrices (pn P') and (pn+1 pn) is ...

Abstract. For any given integer q ≥ 2, we consider sets N of non-negative integers that are defined by linear relations between their q-adic digits (for example, the set of non-negative integers such that the number of 1’s equals twice the number of 0’s in the binary representation). The main goal is to prove that the sequence (αn)n∈N is uniformly distributed modulo 1 for all irrational numbers...

We study the problem of constructing rank-1 lattice rules which have good bounds on the “weighted star discrepancy”. Here the non-negative weights are general weights rather than the product weights considered in most earlier works. In order to show the existence of such good lattice rules, we use an averaging argument, and a similar argument is used later to prove that these lattice rules may ...

Denote by x a random infinite path in the graph of Pascal’s triangle (left and right turns are selected independently with fixed probabilities) and by dn(x) the binomial coefficient at the n’th level along the path x. Then for a dense Gδ set of θ in the unit interval, {dn(x)θ} is almost surely dense but not uniformly distributed modulo 1.

Avigad introduced the notion of UD-randomness based in Weyl’s 1916 definition of uniform distribution modulo one. We prove that there exists a weakly 1-random real that is neither UD-random nor weakly 1-generic. We also show that no 2-generic real can Turing compute a UD-random real. 2000 Mathematics Subject Classification 03D32 (primary); 03D80 (secondary)